<PAGE>   1
================================================================================

                       SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q
(Mark One)

[X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES 
    EXCHANGE ACT OF 1934
For the quarterly period ended September 30, 1998
                                       OR
[ ] TRANSITION PERIOD PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES 
    EXCHANGE ACT OF 1934 
For the transition period from ____________________ to _________________________

                          Commission File Number 1-2475

                                SHELL OIL COMPANY
             (Exact Name of Registrant as Specified in its Charter)

<TABLE>
          <S>                                                        <C>
                         Delaware                                                 13-1299890
                 (State of Incorporation)                            (I.R.S. Employer Identification No.)

             One Shell Plaza, Houston, Texas                                         77002
         (Address of Principal Executive Offices)                                 (Zip Code)
</TABLE>

        Registrant's Telephone Number, Including Area Code (713) 241-6161

Indicate by check mark whether the registrant (1) has filed all reports required
to be filed by Section 13 or 15(d) of the Securities Exchange Act of 1934 during
the preceding 12 months (or for such shorter period that the registrant was
required to file such reports), and (2) has been subject to such filing
requirements for the past 90 days. Yes [X]  No [ ]

The number of shares of Common Stock, $10.00 par value, outstanding as of
September 30, 1998 - 1,000 shares.

                        --------------------------------

                         OMISSION OF CERTAIN INFORMATION

In accordance with General Instruction H of Form 10-Q, the registrant is
omitting Part II, Items 2, 3, and 4 because: 

(1) Royal Dutch Petroleum Company, a Netherlands company, and the "Shell"
Transport and Trading Company, p.l.c., an English company, each of which is a
reporting company under the Securities Exchange Act of 1934 that has filed all
material required to be filed by it pursuant to Section 13, 14, or 15(d)
thereof, own directly or indirectly 60 percent and 40 percent, respectively, of
the shares of the companies of the Royal Dutch/Shell Group of Companies,
including all the equity securities of the registrant; and (2) during the
preceding thirty-six calendar months and any subsequent period of days, there
has not been any material default in the payment of principal, interest, sinking
or purchase fund installment, or any other material default not cured within
thirty days with respect to any indebtedness of the registrant or its
subsidiaries, and there has not been any material default in the payment by the
registrant or its subsidiaries of rentals under material long-term leases.

================================================================================
<PAGE>   2

                          PART I. FINANCIAL INFORMATION

                       SHELL OIL COMPANY AND SUBSIDIARIES
                        CONSOLIDATED STATEMENT OF INCOME
                               Millions of Dollars

<TABLE>
<CAPTION>
                                                        THIRD QUARTER            NINE MONTHS
                                                     -------------------     -------------------
                                                       1998        1997        1998        1997
                                                     -------     -------     -------     -------
<S>                                                  <C>         <C>         <C>         <C>    
REVENUES
    Sales and other operating revenue ............   $ 3,667     $ 7,952     $13,927     $24,253
    Less: Consumer excise and sales taxes ........         2       1,007         798       2,911
                                                     -------     -------     -------     -------
                                                       3,665       6,945      13,129      21,342
    Equity in income of affiliates ...............       146         160         417         391
    Interest and other income ....................       115          32         215         129
                                                     -------     -------     -------     -------
          TOTAL ..................................     3,926       7,137      13,761      21,862
                                                     -------     -------     -------     -------

COSTS AND EXPENSES
    Purchased raw materials and products .........     2,251       4,536       8,249      13,978
    Operating expenses ...........................       639         896       1,931       2,673
    Selling, general and administrative expenses..       147         250         580         750
    Exploration, including exploratory dry holes..        58          62         235         244
    Research expenses ............................        35          45         110         121
    Depreciation, depletion, amortization and
       retirements ...............................       153         442         891       1,418
    Interest and discount amortization ...........       108          52         294         149
    Operating taxes ..............................        54          85         182         278
                                                     -------     -------     -------     -------
          TOTAL ..................................     3,445       6,368      12,472      19,591
                                                     -------     -------     -------     -------

INCOME BEFORE INCOME TAXES AND
MINORITY INTEREST ................................   $   481     $   769     $ 1,289     $ 2,271

    Federal and other income taxes ...............       209         276         483         696
    Minority interest in income
       of subsidiaries ...........................        14          14          60          48
                                                     -------     -------     -------     -------

NET INCOME .......................................   $   258     $   479     $   746     $ 1,527
                                                     =======     =======     =======     =======
</TABLE>

Note: Certain 1997 amounts have been reclassified to conform with current year
presentation.

                          ----------------------------

                         OPERATING SEGMENTS INFORMATION
                               Millions of Dollars

<TABLE>
<CAPTION>
                                                  THIRD QUARTER              NINE MONTHS
                                               --------------------      --------------------
                                                1998         1997         1998         1997
                                               -------      -------      -------      -------
<S>                                            <C>          <C>          <C>          <C>    
SEGMENT NET INCOME (LOSS)
    Oil and Gas Exploration and Production..   $   147      $   268      $   479      $   983
    Downstream Gas .........................        10         --             32         --
    Oil Products ...........................       115          146          243          286
    Chemical Products ......................        67          107          247          364
    Other ..................................        (6)          (1)           4           (3)
    Corporate Items ........................       (75)         (41)        (259)        (103)
                                               -------      -------      -------      -------
NET INCOME .................................   $   258      $   479      $   746      $ 1,527
                                               =======      =======      =======      =======
</TABLE>



                                       2
<PAGE>   3



                       SHELL OIL COMPANY AND SUBSIDIARIES
                           CONSOLIDATED BALANCE SHEET
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                         SEPTEMBER 30,   DECEMBER 31,
                                                              1998           1997 
                                                            -------        -------
<S>                                                         <C>            <C>    
ASSETS                                                                            
CURRENT ASSETS                                                                    
   Cash and cash equivalents ........................       $   385        $   342
   Receivables and prepayments, less allowance for                                
      doubtful accounts .............................         2,721          3,414
   Owing by related parties .........................           740            280
   Inventories of oils and chemicals ................           791            974
   Inventories of materials and supplies ............           147            218
                                                            -------        -------
             TOTAL CURRENT ASSETS ...................         4,784          5,228
INVESTMENTS .........................................        11,840          6,456
LONG-TERM RECEIVABLES AND DEFERRED CHARGES ..........         2,101          1,150
PROPERTY, PLANT AND EQUIPMENT AT COST, LESS                                       
   ACCUMULATED DEPRECIATION, DEPLETION AND                                        
   AMORTIZATION OF $12,422 AT SEPTEMBER 30, 1998                                  
   AND $16,505 AT DECEMBER 31, 1997 .................        12,507         16,767
GOODWILL, NET .......................................         1,016           --  
                                                            -------        -------
             TOTAL ..................................       $32,248        $29,601
                                                            =======        =======
                                                                                  
                                                                                  
LIABILITIES AND SHAREHOLDER'S EQUITY                                              
CURRENT LIABILITIES                                                               
   Accounts payable - trade .........................       $ 1,239        $ 2,257
   Other payables and accruals ......................         1,267          1,281
   Income, operating and consumer taxes .............           102            186
   Owing to related parties .........................         1,146            303
   Short-term debt ..................................         6,293          3,539
                                                            -------        -------
           TOTAL CURRENT LIABILITIES ................        10,047          7,566
LONG-TERM DEBT ......................................           773            585
DEFERRED INCOME TAXES ...............................         3,491          3,339
LONG-TERM LIABILITIES ...............................         2,151          2,154
MINORITY INTEREST ...................................         1,400          1,079
SHAREHOLDER'S EQUITY                                                              
   Common stock - 1,000 shares of $10 per share                                   
       par value ....................................          --             --  
   Capital in excess of par value ...................         2,206          2,206
   Earnings reinvested ..............................        12,180         12,672
                                                            -------        -------
           TOTAL SHAREHOLDER'S EQUITY ...............        14,386         14,878
                                                            -------        -------
           TOTAL ....................................       $32,248        $29,601
                                                            =======        =======
</TABLE>



                                       3

<PAGE>   4



                       SHELL OIL COMPANY AND SUBSIDIARIES
                      CONSOLIDATED STATEMENT OF CASH FLOWS
                               Millions of Dollars

<TABLE>
<CAPTION>
                                                                       NINE  MONTHS
                                                                   --------------------
                                                                    1998         1997
                                                                   -------      -------
<S>                                                                <C>          <C>    
CASH FLOWS FROM OPERATING ACTIVITIES
   Net Income ...................................................  $   746      $ 1,527
   Adjustments to reconcile net income to net cash
       provided by operating activities:
         Depreciation, depletion, amortization and retirements...      891        1,418
         Dividends in excess of (less than) equity income .......       17         (221)
         (Increases) decreases in working capital:
              Receivables and prepayments .......................     (549)         910
              Inventories .......................................     (139)        (246)
              Current payables and accruals .....................      446         (920)
           Deferred income taxes ................................      (93)         228
           Minority interest in income of subsidiaries ..........       60           48
           Other noncurrent items ...............................      190         (232)
                                                                   -------      -------
              Net Cash Provided by Operating Activities .........    1,569        2,512
CASH FLOWS PROVIDED BY (USED FOR) INVESTING ACTIVITIES
   Capital expenditures:
      Acquisition of Tejas ......................................   (1,376)          --
      Other .....................................................   (1,729)      (2,244)
   Proceeds from property sales and salvage .....................      160          176
   Other investments ............................................      507          (16)
                                                                   -------      -------
              Net Cash Used for Investing Activities ............   (2,438)      (2,084)
                                                                   -------      -------
CASH FLOWS PROVIDED BY (USED FOR) FINANCING ACTIVITIES
   Proceeds from issuance of long-term debt .....................      324          211
   Principal payments on long-term debt .........................     (332)        (484)
   Proceeds from sales of securities of subsidiaries ............      321           32
   Dividends to shareholder .....................................   (1,238)      (1,200)
   Dividends to minority interest ...............................      (60)         (43)
   Increase (decrease) in short-term obligations ................    1,897        1,183
                                                                   -------      -------
              Net Cash Provided by Financing Activities .........      912         (301)
                                                                   -------      -------
NET CASH FLOWS
   Increase (Decrease) in cash and cash equivalents ..........     $    43      $   127
                                                                   =======      =======
CASH AND CASH EQUIVALENTS
   Balance at beginning of period ............................     $   342      $   393
   Increase (decrease) in cash and cash equivalents ..........          43          127
                                                                   -------      -------
              Balance at end of period .......................     $   385      $   520
                                                                   =======      =======
</TABLE>



                                       4

<PAGE>   5


                       SHELL OIL COMPANY AND SUBSIDIARIES

                      NOTES TO INTERIM FINANCIAL STATEMENTS

A.  INTERIM FINANCIAL STATEMENT MATTERS

         The unaudited financial statements and summarized notes of Shell Oil
Company ("the Company") and its consolidated subsidiaries ("Shell Oil") included
in this report do not include complete financial information and should be read
in conjunction with the Consolidated Financial Statements and the Notes to
Consolidated Financial Statements filed with the Securities and Exchange
Commission ("the Commission") in the Registrant's Annual Report on Form 10-K for
the year ended December 31, 1997. The financial information presented in the
financial statements included in this report reflects all adjustments which are,
in the opinion of management, necessary for a fair statement of results for the
interim periods presented. Any such adjustments are of a normal recurring
nature, except as may otherwise be described in Management's Discussion and
Analysis of Financial Condition and Results of Operations.

         The results for the third quarter and first nine months of 1998 should
not be construed as necessarily indicative of future financial results.


B.  SIGNIFICANT 1998 ALLIANCES AND ACQUISITIONS

         EQUILON ENTERPRISES LLC. On January 15, 1998, Shell Oil and Texaco Inc.
("Texaco") reached agreement on the formation and operational start up,
effective January 1, 1998, of Equilon Enterprises LLC ("Equilon"). Equilon is a
joint venture which combines major elements of both companies' western and
midwestern United States refining and marketing businesses and both companies'
nationwide trading, transportation and lubricants businesses. Shell Oil owns 56
percent of Equilon but does not exercise control and therefore accounts for its
investment in Equilon using the equity method of accounting. Shell Oil recorded
its investment in Equilon by removing from its consolidated balance sheet the
values of the assets and liabilities it contributed to the joint venture, or
approximately $6.2 billion and $2.3 billion, respectively, and, in turn,
recording the net of these amounts, or approximately $3.9 billion as its equity
investment in Equilon. Further detail concerning this new venture was included
in the Company's Current Report on Form 8-K filed with the Commission on January
30, 1998.

         TEJAS GAS CORPORATION. In January 1998 Shell Oil acquired all of the
outstanding common stock of Tejas Gas Corporation ("Tejas"), a natural gas
pipeline company engaged in the business of purchasing, gathering, processing,
treating, storing, transporting and marketing natural gas, for $61.50 per share
which, on a fully diluted common stock basis, represented an aggregate common
stock purchase price of approximately $1.4 billion. In addition, Shell Oil
assumed Tejas' balance sheet debt and preferred stock of approximately $1.4
billion. Shell Oil accounted for this transaction using the purchase method of
accounting. Prior to this transaction, Shell Oil, Tejas and Shell Canada jointly
owned Coral Energy, L.P. ("Coral"), a gas marketing enterprise, with an
ownership interest of 44 percent, 44 percent and 12 percent, respectively. Shell
Oil accounted for its 44 percent interest in Coral using the equity method of
accounting; however, with the completion of the Tejas acquisition, Shell Oil
fully consolidates its now 88 percent ownership interest in Coral.



                                       5
<PAGE>   6




         MOTIVA ENTERPRISES LLC. As reported in the Company's Current Report on
Form 8-K filed with the Commission on July 1, 1998, on July 1, 1998 Shell Oil,
Texaco and Saudi Arabian Oil Company (Saudi Aramco) jointly announced the
formation and operational start-up of Motiva Enterprises LLC (Motiva), a joint
venture combining major elements of the three companies' eastern and Gulf Coast
United States refining and marketing businesses, including assets previously
held by Star Enterprise, a partnership of corporate affiliates of Texaco and
Saudi Aramco. Shell Oil has 35 percent ownership of Motiva, and Texaco and Saudi
Refining, Inc., a corporate affiliate of Saudi Aramco, each have 32.5 percent
ownership of the company (such ownership to be subject to adjustment in the
future based on the performance of the assets). Shell Oil accounts for its
investment in Motiva using the equity method of accounting. Shell Oil recorded
its investment in Motiva by removing from its consolidated balance sheet the
values of the assets and liabilities it contributed to the joint venture, or
approximately $2.2 billion and $0.5 billion, respectively, and, in turn,
recording the net of these amounts, or approximately $1.7 billion as its equity
investment in Motiva.

         The following summary, prepared on a pro forma basis, presents the
Shell Oil results of operations for the three month and nine month periods
ending September 30, 1997 and the nine month period ending September 30, 1998 as
if Equilon and Motiva had been formed and Tejas had been acquired on January 1,
1997:

                       SHELL OIL COMPANY AND SUBSIDIARIES
                  PRO FORMA CONSOLIDATED RESULTS OF OPERATIONS

<TABLE>
<CAPTION>
                             Three Months Ended                Nine Months Ended             Nine Months Ended
                             September 30, 1997               September 30, 1997            September 30, 1998
                             ------------------               -------------------           ------------------
                                                             (Millions of dollars)
     <S>                           <C>                             <C>                           <C>
     Gross Revenues........        $ 3,405                         $ 12,287                      $  10,232
     Net income............            454                            1,474                            740
</TABLE>

C.  EQUILON ENTERPRISES LLC
         The following unaudited financial information for Equilon is reflected
on a 100 percent Equilon basis:

<TABLE>
<CAPTION>
                                                            Three Months Ended               Nine Months Ended
                                                            September 30, 1998              September 30, 1998
                                                            ------------------              ------------------
                                                                          (Millions of dollars)
      <S>                                                     <C>                                 <C>
      Gross Revenues...................................       $   6,100                           18,195
      Income Before Tax................................             232                              542
</TABLE>

         As a limited liability company, Equilon's results of operations do not
include income tax liability, but rather the income tax liability is reflected
in the results of operations of the parent companies. Shell Oil's 56 percent
equity share of Equilon's Income Before Tax and the corresponding income tax
expense is reflected currently in Shell Oil's Consolidated Statement of Income.



                                       6
<PAGE>   7

D.  MOTIVA ENTERPRISES LLC

         The following unaudited financial information for Motiva is reflected
on a 100 percent Motiva basis:

<TABLE>
<CAPTION>
                                                          Three Months Ended
                                                          September 30, 1998
                                                          ------------------
                                                         (Millions of dollars)
      <S>                                                     <C>
      Gross Revenues...................................       $   2,877
      Income Before Tax................................              59
</TABLE>

         As a limited liability company, Motiva's results of operations do not
include income tax liability, but rather the income tax liability is reflected
in the results of operations of the parent companies. Shell Oil's 35 percent
equity share of Motiva's Income Before Tax and the corresponding income tax
expense is reflected currently in Shell Oil's Consolidated Statement of Income.


E.  CONTINGENCIES AND OTHER MATTERS

         Shell Oil is subject to a number of possible loss contingencies. These
include actions based upon environmental laws involving present and past
operating and waste disposal locations and related private claims, contract and
product liability actions and federal, state and private actions challenging the
correctness of oil and gas royalty calculations. In addition, federal, state and
local income, property and excise tax returns are being examined and certain
interpretations by Shell Oil of complex tax statutes, regulations and practices
are being challenged.

         Since 1984, the Company has been named with others as a defendant in
numerous product liability cases, including class actions, involving the failure
of residential plumbing systems constructed with polybutylene plastic pipe. The
Company has also been sued regarding failures in polybutylene pipe connecting
users with utility water lines and polybutylene pipe used in municipal water
distribution systems. The Company fabricated the resin for this pipe. Two other
substantial manufacturers made the resins for the polyacetal insert fittings
used in many of the residential plumbing systems (the fittings co-defendants)
and are also defendants in those cases. The Company and the fittings
co-defendants have agreed on a mechanism to fund the payment of most of the
residential plumbing claims in the United States as the result of two class
action settlements (the "class action settlement"). The class action settlement
provides for the creation of an entity to receive and handle claims and for a
$950 million fund to pay such claims, which claims may be filed until 2009,
depending on various factors. If the settlement funds are exhausted, additional
funds may be provided by the defendants, or claimants who have not received
their full benefits under the class action settlements may seek their remedy in
a new court proceeding at that time. One fittings co-defendant has agreed to
fund 10% of all acetal fittings costs related to the class action settlement;
the Company and the other fittings co-defendant have agreed to arbitration to
determine how the remaining acetyl fittings portion of the costs will be shared
between them. Additionally, claims continue to be filed involving problems with
polybutylene pipe used in municipal water distribution systems. The Company will
continue to defend these matters vigorously but it cannot currently predict when
or how polybutylene related matters will finally be resolved.

         In an October 1997 decision by the United States District Court in
Delaware, certain income tax credits recorded by Shell Oil in previous years
arising out of production of oil from tar sands were denied because the Court
determined that Shell Oil used the wrong definition of tar sands production to
calculate the same. Shell Oil is currently examining the effect of this decision
on other previously recorded tar 




                                       7
<PAGE>   8
sands tax credits. However, Shell Oil believes that the District Court decision
was incorrect and intends to vigorously appeal such decision. In any case, Shell
Oil believes that many of its tar sands tax credits are validly claimed under
the alternative definition asserted by the government in the District Court
case.

         The Company's assessment of these matters is continuing. Future
provisions may be required as administrative and judicial proceedings progress
and the scope and nature of remediation programs and related costs estimates are
clarified. However, while periodic results may be significantly affected by
costs in excess of provisions related to one or more of these proceedings, based
upon developments to date, the management of the Company anticipates that it
will be able to meet related obligations without a material adverse effect on
its financial position.

                            ------------------------


                     MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                  FINANCIAL CONDITION AND RESULTS OF OPERATIONS


RESULTS OF OPERATIONS

         Shell Oil Company reported third quarter net income of $258 million, a
decrease of $221 million from the same quarter of 1997. Excluding special items
in both quarters, adjusted net income in the third quarter of 1998 totaled $188
million, a decrease of $271 million from the 1997 quarter.

         Excluding special items, the key factor contributing to the lower
income in the third quarter of 1998 compared to the same period in 1997 was the
significantly lower price for crude oil. Refined product margins as well as
chemical margins also declined, reflecting the competitive nature of these
businesses during a period of economic slowdown. These declines more than offset
the benefits achieved from increased production of domestic crude oil.

         Net income for the nine months of 1998 totaled $746 million, decreasing
$781 million from the comparable 1997 period. Excluding special items in both
periods, adjusted net income for the nine months of 1998 totaled $671 million,
decreasing $763 million or 53 percent.

         Contributing to the earnings decline in the first nine months of 1998
were lower prices for crude oil and natural gas, more than offsetting
significant production increases of domestic crude oil and natural gas.

         Special items in the 1998 periods increased net income $70 million for
the quarter and $75 million for the nine months. In comparison, special items in
1997 increased net income by $20 million for the quarter and $93 million in the
first nine months. Special items in the 1998 quarter included gains from the
sale of exploration and production assets including a $120 million benefit from
the sale of assets in Yemen, which on a pretax basis, is reflected as a
reduction of $201 million in Depreciation, depletion, amortization and
retirements on the Consolidated Statement of Income. Partially offsetting these
gains were charges to provisions for plant closures, severance provisions and
litigation settlements.

         During 1998, crude oil, and to a lesser extent natural gas, prices 
have declined sharply, and our current analysis of the foreseeable future 
supports a prolonged period of lower prices. This analysis has led to a review 
of our planning and operational premises and has implications regarding our 
existing asset


                                       8
<PAGE>   9
portfolio. While analysis is ongoing and no decisions can be made until such 
analysis is complete, it is possible that further asset restructuring may occur 
and that, in the fourth quarter of 1998, a charge could be incurred as asset 
carrying values are re-examined in the light of the changed outlook.

OIL AND GAS EXPLORATION AND PRODUCTION

<TABLE>
<CAPTION>
Income Highlights                                                Third Quarter                     Nine Months
-----------------                                           ----------------------          -----------------------
(millions of dollars)                                        1998            1997            1998             1997
                                                            ------          ------          ------           ------
   <S>                                                      <C>             <C>              <C>             <C>   
   Segment Net Income..................................     $  147          $  268           $ 479           $  983
   Special Items ......................................         56              10              98               46
                                                            ------          ------           -----            -----
   Adjusted Net Income.................................         91             258             381              937
</TABLE>

         Oil and gas exploration and production earnings in the third quarter of
1998 totaled $147 million, decreasing $121 million from 1997. For the nine
months of 1998, earnings were $479 million, down $504 million. Excluding special
items in the comparable periods, adjusted net income was down $167 million in
the 1998 quarter versus 1997, and down $556 million in the nine-month
comparison.

         Lower prices for crude oil and natural gas during the third quarter and
the first nine months of 1998 compared to both 1997 periods more than offset the
benefit from increased domestic crude oil production. For the third quarter of
1998, Shell's domestic crude oil prices averaged $10.72 per barrel, down $6.05
per barrel, while natural gas prices averaged $1.88 per thousand cubic feet,
down $.40 from the same 1997 period. In the first nine months of 1998, Shell's
average domestic crude oil prices were 36 percent lower, and natural gas prices
were 16 percent lower than in the same 1997 periods.

         Domestic production of crude oil in the third quarter of 1998 averaged
431,000 barrels per day, an increase of 18,000 barrels per day over the same
period in 1997. For the nine months of 1998, domestic crude oil production
averaged 447,000 barrels per day, up 44,000. Natural gas production was down 4
percent in the quarter comparison, but up 8 percent in the nine months of 1998
over 1997. New and increased crude oil and natural gas production in the
deepwater Gulf of Mexico was partially offset by normal declines elsewhere.
Third quarter 1998 crude oil and natural gas production was affected by
temporary curtailments from weather conditions due to hurricanes in the Gulf of
Mexico, while natural gas production was also affected by maintenance.

         Domestic crude oil and natural gas production numbers include Shell
Oil's net production plus a pro rata share, based on ownership interest, of
domestic equity companies' production; price and expenditure information
excludes equity company data. Equity companies are those companies in which
Shell Oil has significant influence but not control.


DOWNSTREAM GAS

<TABLE>
<CAPTION>
Income Highlights                                                Third Quarter                     Nine Months
-----------------                                            ---------------------          -----------------------
(millions of dollars)                                         1998           1997            1998             1997
                                                             ------        -------          ------           ------

   <S>                                                       <C>           <C>              <C>              <C>   
   Segment Net Income..................................      $  10              --          $   32               --
   Special Items ......................................         --              --              --               --
                                                             -----         -------          ------           ------
   Adjusted Net Income.................................         10              --              32               --
</TABLE>





                                       9
<PAGE>   10
         Downstream gas, a new operating segment of Shell Oil, had earnings for
the third quarter of 1998 of $10 million. For the first nine months of 1998,
earnings totaled $32 million. This is the initial year of reporting, and as a
result no comparative earnings data are available for 1997.

         In January 1998, Shell Oil acquired Tejas, including Tejas' interest in
Coral, as further discussed in Note B of the Notes to Interim Financial
Statements. In addition to Shell Oil's previously existing natural gas marketing
business and its infrastructure of natural gas pipelines in the Gulf of Mexico,
the new downstream gas segment also includes the operations of Tejas, Coral and
Corpus Christi Natural Gas, which was acquired in 1997.

         During the third quarter, transported natural gas volumes were about
6,237 million cubic feet per day, down about 9 percent from the second quarter
of this year. Gas transportation margins, however, were up slightly from the
prior quarter. Gas processing throughputs were about 76,000 barrels per day,
down almost 30 percent from last quarter, and gas processing margins declined
sharply, both due to adverse market conditions.

         Phase One of the Destin Pipeline commenced operation in September, with
throughput of about 100 million cubic feet per day.

OIL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                                               Third Quarter                     Nine Months
-----------------                                           ----------------------          -----------------------
(millions of dollars)                                        1998            1997            1998             1997
                                                            ------          ------          ------           ------
   <S>                                                      <C>             <C>             <C>              <C>
   Segment Net Income..................................     $  115          $  146          $  243           $  286
   Special Items ......................................         14               8             (16)               3
                                                            ------          ------          ------            -----
   Adjusted Net Income.................................        101             138             259              283
</TABLE>

         Shell Oil's oil products segment consists primarily of its share of
results from Equilon Enterprises LLC (Shell Oil interest - 56%), Motiva
Enterprises LLC, which began operations in July (Shell Oil interest - 35%), and
the Deer Park Refining Limited Partnership (Shell Oil interest - 50%). Shell Oil
accounts for its investment in each of these companies using the equity method
of accounting.

         Oil products earnings totaled $115 million in the third quarter of
1998, a decrease of $31 million from 1997. In the first nine months of 1998,
earnings totaled $243 million, down $43 million from 1997. Excluding special
items in comparable periods, adjusted net income decreased $37 million versus
the 1997 quarter, and $24 million in the nine-month comparison.

         For the third quarter and the first nine months of 1998, refined
product margins declined compared to the same periods in 1997. Refining margins
fell sharply during the third quarter of 1998, only partially offset by improved
marketing margins.

         As discussed in Note B of the Notes to Interim Financial Statements,
operations began, effective January 1, 1998, in Equilon, the new refining and
marketing venture jointly owned by Shell Oil and Texaco. Equilon combines major
elements of both companies' western and midwestern United States refining and
marketing businesses and their nationwide trading, transportation and lubricants
businesses.

         Also as discussed in Note B of the Notes to Interim Financial
Statements, effective July 1, 1998, 



                                       10
<PAGE>   11
operations began in Motiva, a joint venture between Shell Oil, Texaco and Saudi
Refining, Inc. Motiva combines major elements of the three companies' eastern
and Gulf Coast U.S. refining and marketing businesses. 

CHEMICAL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                                                Third Quarter                    Nine Months
-----------------                                            ---------------------          -----------------------
(millions of dollars)                                         1998           1997            1998             1997
                                                             ------         ------          ------           ------
   <S>                                                       <C>            <C>             <C>              <C>   
   Segment Net Income..................................      $  67          $  107          $  247           $  364
   Special Items ......................................         --               2              (9)              (6)
                                                             -----          ------          ------           ------
   Adjusted Net Income.................................         67             105             256              370
</TABLE>

         Chemical products earnings in the third quarter of 1998 were $67
million, a decrease of $40 million from 1997. For the nine months of 1998,
chemical products earnings totaled $247 million, down $117 million. Excluding
special items in the comparable periods, adjusted net income was down $38
million in the 1998 quarter and $114 million in the nine months of 1998.

         Income declined in both 1998 periods compared to 1997 due to lower
margins for primary chemicals, resulting from significant price declines for
olefins. As the result of Hurricane Georges in September, receipts of feedstocks
were temporarily interrupted resulting in drawdowns of higher priced
inventories. Sales volumes of primary chemicals and downstream products
increased, with higher sales of aromatics, olefins and solvents in both 1998
periods over 1997.

OTHER

         The other segment incurred a net loss in the third quarter of 1998 of
$6 million compared to a net loss of $1 million in same 1997 quarter. For the
nine months of 1998, the other segment had net income of $4 million compared to
a net loss of $3 million in the comparable 1997 period.

CORPORATE ITEMS

         Corporate charges totaled $75 million and $259 million respectively in
the third quarter and first nine months of 1998, compared to charges of $41
million and $103 million, respectively, in the comparable 1997 periods.
Excluding special items, corporate charges in the 1998 nine month period were
$261 million compared to charges of $153 million in the 1997 period. There were
no special items in the 1998 and 1997 third quarters. Higher financing costs in
both 1998 periods were the primary factors in the increases.

FINANCIAL CONDITION

CAPITAL RESOURCES AND LIQUIDITY

         Cash flow provided by operating activities totaled $1,569 million for
the first nine months of 1998, compared with $2,512 million in the comparable
period last year, a decrease of $943 million. The period to period decrease was
attributable to lower earnings. Cash generated from operating activities,
coupled with 



                                       11

<PAGE>   12
an increase in debt and sale of securities totaling $3,779 million, and proceeds
from property sales and other investments of $667 million in the first nine
months of 1998 were used primarily for capital expenditures of $3,105 million
and dividend payments of $1,298 million.

OTHER MATTERS

YEAR 2000 ISSUES

         Shell Oil has been engaged in the identification, analysis and
remediation of the Year 2000 computer date ("Y2K") problem since 1997. This
problem is associated with the incorrect interpretation of computer stored dates
which are numerically related to the change of the millennium. Large and small
computing devices and associated software used in running large business
applications as well as some kinds of industrial automation systems are
potentially subject to this problem.

         Under the Shell Oil governance model, each business organization takes
independent responsibility for identification, analysis and remediation of Y2K
problems within such business. Year 2000 Teams have been formed in each business
organization to deal with the Y2K problem. The Teams communicate through a
central focal point committee which tracks and updates Year 2000 progress
reports from each business at the overall Shell Oil level. This Shell Oil Year
2000 Team also provides a vehicle for exchange of ideas and experiences to
achieve improved efficiencies and results. The Chief Information Officer of each
business represents the organization on the Shell Oil Year 2000 Team.
Confirmation is sought from those equity companies where Shell Oil has a
significant voting interest of an active and appropriate Y2K program.

         Shell Oil Year 2000 Teams have identified and are in the process of
analyzing the use of computer technology in the respective businesses to
determine where there may be business operation issues associated with the Y2K
problem. Priority in such review is placed first on health, safety and
environmental issues, then on areas which may materially affect third parties or
revenues. This review necessarily includes both internal uses of technology as
well as key trading interfaces which might be adversely affected by third party
customers', suppliers' or service providers' use of computing technology with
Y2K problems.

         The Shell Oil businesses have divided their analysis of the Y2K problem
into three areas: 1) business information, computing and telecommunication
systems ("IT Systems"); 2) industrial automation systems ("Imbedded Technology
Systems"); and 3) potential problems involving the Y2K problems of third party
suppliers, customers and service providers (each a "Third Party" and together
"Third Parties"). The internal portion of this analysis and inventory of
potential problems has been completed in all areas believed to be associated
with critical IT Systems. Final stages of the inventory of non-critical IT
Systems, Imbedded Technology Systems and Third Party interfaces are in progress
and are expected to be completed by January 1, 1999. Remediation of Y2K problems
in all problem areas which have been identified is scheduled to be accomplished
by the end of June 1999. Remediation is primarily handled by internal company
staff, although some outside companies are involved in the remediation
associated with particular systems. It is expected that analysis of interfaces
with key Third Parties will continue into August of 1999 to insure appropriate
coordination and, where necessary or appropriate, development of contingency
plans in the event a Third Party cannot meet its commitments to Shell Oil.

         Due to a business-driven demand for improved computing systems
throughout Shell Oil, the primary business computing systems in each Shell Oil
business organization are being replaced in this 



                                       12

<PAGE>   13
decade. These new systems are represented to be fully Y2K compliant by their
manufacturers and Shell Oil believes that it will have no Y2K issues in its
important business computing systems. While acquisition of these new IT Systems
for business computing avoided the necessity of dealing with Y2K issues in the
predecessor systems, the cost of these systems was incurred primarily for other
business reasons. The majority of the costs of each such system are being
capitalized and charged against income over the estimated useful life of the
systems. Funds for the acquisition of these systems are included in the capital
budget of the respective businesses.

         Costs associated with the identification, analysis and remediation of
Y2K problems, including the repair or replacement of imbedded technology
systems, are currently projected by the businesses at approximately $150
million. These costs are being charged to expense as incurred. At of the end of
the third quarter of 1998, approximately one half of such expenditure has been
committed or incurred. Funding for these efforts has been a part of the normal
budgeting of current operations within each business.

         As to the Imbedded Technology Systems used in industrial automation
within Shell Oil, the businesses are in the process of inventorying, analyzing
and testing these systems. The various Shell Oil businesses have used several
approaches to the identification and analysis of the potential Y2K problem in
such systems. Identification of imbedded technology issues is handled by field
installation inspection in some cases and in other cases by inspection of design
blueprints, with field testing to confirm the adequacy of design information.
Inventory and analysis of potential problems in this area are expected to be
completed by January 1, 1999 with all remediation completed by July 1, 1999.

         Beyond the internal programs in the Shell Oil businesses, significant
efforts are underway to understand the position of key Third Parties with
respect to possible business interruptions due to Y2K problems in their systems.
Current indications are that all of Shell Oil's major Third Party customers,
suppliers and service providers are well aware of this problem and have programs
underway to address it. Discussions will continue with these Third Parties
during 1998 and 1999 to best position Shell Oil to continue its business
operations without interruption. Specific attention is being given to Third
Parties that have critical relationships with Shell Oil to attempt to insure
that there is not a disruption in the flow of those products or services which
are essential to the ability of Shell Oil to carry on its business operations.
Since Shell Oil will remain dependent on Third Party confirmations in certain
areas, appropriate contingency plans will be considered where viable.

         The key objective of Shell Oil's extensive program to address the Y2K
computer technology problem is the significant reduction or complete elimination
of the risk to business operations. Internal audits are planned to occur by the
end of the first quarter of 1999 in all of Shell Oil's major business units to
further validate the coverage of the Y2K remediation programs. As a contingency
plan, steps will be taken to manage this issue around key dates where
unanticipated Y2K problems may arise. Such plans include the positioning of
skilled computer staff in the various areas of business operations to insure
quick response to any unexpected Y2K problems arising due to failure to
previously detect the problem or due to inadequate remediation. Shell Oil plans
to review its current business operation contingency plans in the first half of
1999 to understand what additions may be appropriate to those plans to deal with
residual risk associated with Y2K problems. This will include decisions on
operating schedules and staff deployment during time periods in which
Y2K-related problems are most likely to occur.

         Shell Oil realizes that its failure or the failure of critical Third
Parties to identify and correct a material Y2K problem could result in an
interruption in or failure of certain normal business activities or operations.
While such failures could materially and adversely affect Shell Oil's operations
and relationships with Third Parties, the objective of Shell Oil's Year 2000
Teams is the significant reduction 



                                       13
<PAGE>   14
or complete elimination of these risks. Shell Oil expects to achieve this goal.
As stated above, steps will be taken to manage this issue around key dates where
the problem may arise.

         The discussion of Shell Oil's Y2K efforts and expectations regarding
the same includes forward-looking statements within the meaning of Section 27A
of the Securities Act of 1933. Shell Oil's ability to achieve Year 2000
readiness and the level of costs associated therewith could be adversely
impacted by, among other things: the availability and cost of programming and
testing resources; vendors' ability to install or modify hardware, software or
other computer technology in Shell Oil systems; and unanticipated problems
identified as the Year 2000 Teams conclude their analysis and verification of IT
and Imbedded Technology Systems. Additionally, no precedent exists as to the
manner in which to fully detect and eliminate Y2K risks. Thus, it is possible
that despite all efforts to identify, analyze and remediate Y2K related
problems, not all potential problems may be detected or all remediation efforts
operate as intended; it is impossible to accurately predict the impact on Shell
Oil in any such case. Finally, the failure of Third Parties to achieve
acceptable Y2K compliance and the absence of available contingent suppliers and
resources could also materially adversely affect Shell Oil.


STATEMENTS OF FINANCIAL ACCOUNTING STANDARDS NOT YET ADOPTED

         In June 1998 the Financial Accounting Standards Board issued Statement
of Financial Accounting Standards No. 133, "Accounting for Derivative
Instruments and Hedging Activities" ("SFAS 133"). This new standard is effective
for fiscal years beginning after June 15, 1999 (January 1, 2000 for the
Company). SFAS 133 requires that all derivative instruments be recorded on the
balance sheet at their fair value. Changes in the fair value of derivatives are
recorded each period in current earnings or other comprehensive income,
depending on whether a derivative is designated as part of a hedge transaction
and, if it is, the type of hedge transaction. The Company has not yet completed
its evaluation of the impact of the adoption of this new standard.


         In addition to the economic conditions and other matters discussed
above affecting Shell Oil, the operations, earnings and financial condition of
Shell Oil may be affected by political developments; litigation; and
legislation, regulation and other actions taken by federal, state, local and
foreign governmental entities, including those matters discussed in Note D of
the Notes to Interim Financial Statements.

                            -------------------------



                                       14
<PAGE>   15



                           PART II. OTHER INFORMATION


ITEM 1.  LEGAL PROCEEDINGS.

         As previously reported on the Company's Annual Report on Form 10-K for
the year ended December 31, 1997, the Company and certain of its affiliates were
defendants in contract litigation involving the polyolefins business brought
against them in 1995 by Union Carbide in the United States District Court for
the Southern District of New York. On July 31, 1998 Shell Oil Company and the
other Shell defendants settled the lawsuit. Under the settlement agreement, the
Shell defendants paid a sum of money to Union Carbide, without an admission of
guilt; the settlement agreement includes a general release of all claims.


     During the third quarter, a Shell subsidiary paid $137,500 in fines to
settle certain alleged permit violations. The allegations involved effluent
discharges in excess of OCS permit limitations

ITEM 6.   EXHIBITS AND REPORTS ON FORM 8-K

         (a) Exhibits.

             27.  Financial Data Schedule.

         (b) Reports on Form 8-K.
             During the third quarter of 1998, Shell Oil filed the following
             Current Report on Form 8-K:

             July 1, 1998 (Date of Earliest event reported: July 1, 1998)
             Item 5. Other Events - reported the formation and operational
             startup of Motiva Enterprises LLC on July 1, 1998.



                                   SIGNATURES

         Pursuant to the requirements of the Securities Exchange Act of 1934,
the Registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.


                                                  SHELL OIL COMPANY



                                        By         N. J. CARUSO
                                           -----------------------------------
                                               N. J. Caruso, Controller
                                               (Principal Accounting and
                                               Duly Authorized Officer)


Date:  November 12, 1998




                                       15
<PAGE>   16



                                INDEX TO EXHIBITS


<TABLE>
<CAPTION>
Exhibit
Number          Description
-------         -----------
  <S>           <C>
  27            Financial Data Schedule
</TABLE>