<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 June 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)

               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)
        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 June
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>
                                                                 SECOND QUARTER            SIX MONTHS
                                                               -------------------     --------------------
                                                                 1998        1997        1998        1997
                                                               -------     -------     -------     --------
<S>                                                            <C>         <C>         <C>         <C>    
REVENUES
    Sales and other operating revenue ....................     $ 5,235     $ 7,908     $10,261     $16,301
    Less:  Consumer excise and sales taxes ...............         419         988         796       1,904
                                                               -------     -------     -------     -------
                                                                 4,816       6,920       9,465      14,397
    Equity in income of affiliates .......................         151         135         271         231
    Interest and other income ............................          63          38          99          97
                                                               -------     -------     -------     -------
          TOTAL ..........................................       5,030       7,093       9,835      14,725
                                                               -------     -------     -------     -------

COSTS AND EXPENSES
    Purchased raw materials and products .................       3,056       4,305       6,000       9,440
    Operating expenses ...................................         613       1,049       1,292       1,777
    Selling, general and administrative expenses..........         214         287         433         500
    Exploration, including exploratory dry holes .........         107          84         176         162
    Research expenses ....................................          38          40          75          76
    Depreciation, depletion, amortization and
       retirements .......................................         342         478         738         977
    Interest and discount amortization ...................         101          52         186          97
    Operating taxes ......................................          60          80         128         193
                                                               -------     -------     -------     -------
          TOTAL ..........................................       4,531       6,375       9,028      13,222
                                                               -------     -------     -------     -------

INCOME BEFORE INCOME TAXES AND
MINORITY INTEREST ........................................     $   499     $   718     $   807     $ 1,503

    Federal and other income taxes .......................         162         167         274         420
    Minority interest in income
       of subsidiaries ...................................          21          20          45          35
                                                               -------     -------     -------     -------

NET INCOME ...............................................     $   316     $   531     $   488     $ 1,048
                                                               =======     =======     =======     =======
</TABLE>

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

                         OPERATING SEGMENTS INFORMATION
                               Millions of Dollars

<TABLE>
<CAPTION>
                                                  SECOND QUARTER            SIX MONTHS
                                               -------------------      ------------------
                                                1998         1997        1998        1997
                                               ------       ------      ------      ------
<S>                                            <C>          <C>         <C>         <C>   
SEGMENT NET INCOME (LOSS)
    Oil and Gas Exploration and Production     $  184       $  262      $  332      $  715
    Downstream Gas .......................         14           --          23          --
    Oil Products .........................        117          128         128         139
    Chemical Products ....................         75          151         180         257
    Other ................................          5           (2)          9          (1)
    Corporate Items ......................        (79)          (8)       (184)        (62)
                                               ------       ------      ------      ------
NET INCOME ...............................     $  316       $  531      $  488      $1,048
                                               ======       ======      ======      ======
</TABLE>

2

<PAGE>   3

                       SHELL OIL COMPANY AND SUBSIDIARIES
                           CONSOLIDATED BALANCE SHEET
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                       JUNE 30,    DECEMBER 31,
                                                         1998         1997
                                                       --------    ------------
<S>                                                    <C>           <C>    
ASSETS
CURRENT ASSETS
   Cash and cash equivalents .....................     $   286       $   342
   Receivables and prepayments, less allowance for
      doubtful accounts ..........................       3,056         3,414
   Owing by related parties ......................         562           280
   Inventories of oils and chemicals .............         902           974
   Inventories of materials and supplies .........         176           218
                                                       -------       -------
             TOTAL CURRENT ASSETS ................       4,982         5,228
INVESTMENTS ......................................      10,289         6,456
LONG-TERM RECEIVABLES AND DEFERRED CHARGES .......       1,885         1,150
PROPERTY, PLANT AND EQUIPMENT AT COST, LESS
   ACCUMULATED DEPRECIATION, DEPLETION AND
   AMORTIZATION OF $13,413 AT JUNE 30, 1998
   AND $16,505 AT DECEMBER 31, 1997 ..............      14,184        16,767
GOODWILL, NET ....................................       1,022            --
                                                       -------       -------
             TOTAL ...............................     $32,362       $29,601
                                                       =======       =======


LIABILITIES AND SHAREHOLDER'S EQUITY
CURRENT LIABILITIES
   Accounts payable - trade ......................     $ 1,541       $ 2,257
   Other payables and accruals ...................       1,366         1,281
   Income, operating and consumer taxes ..........          81           186
   Owing to related parties ......................         394           303
   Short-term debt ...............................       6,756         3,539
                                                       -------       -------
           TOTAL CURRENT LIABILITIES .............      10,138         7,566
LONG-TERM DEBT ...................................         769           585
DEFERRED INCOME TAXES ............................       3,338         3,339
LONG-TERM LIABILITIES ............................       2,252         2,154
MINORITY INTEREST ................................       1,324         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 ...........................      12,335        12,672
                                                       -------       -------
           TOTAL SHAREHOLDER'S EQUITY ............      14,541        14,878
                                                       -------       -------
           TOTAL .................................     $32,362       $29,601
                                                       =======       =======
</TABLE>


3

<PAGE>   4


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

<TABLE>
<CAPTION>
                                                                                       SIX MONTHS
                                                                                 ----------------------
                                                                                   1998          1997
                                                                                 --------      --------
<S>                                                                              <C>           <C>
CASH FLOWS FROM OPERATING ACTIVITIES
   Net Income ................................................................  $   488        $  1,048
   Adjustments to reconcile net income to net cash
       provided by operating activities:
      Depreciation, depletion, amortization and retirements ..................      738             977
           Dividends less than equity income .................................      (76)           (152)
           (Increases) decreases in working capital:
              Receivables and prepayments ....................................     (202)            907
              Inventories ....................................................     (161)           (200)
              Current payables and accruals ..................................     (295)         (1,098)
           Deferred income taxes .............................................     (250)            142
           Minority interest in income of subsidiaries .......................       45              35
           Other noncurrent items ............................................      225            (142)
                                                                                -------        --------
           Net Cash Provided by Operating Activities .........................      512           1,517
CASH FLOWS PROVIDED BY (USED FOR) INVESTING ACTIVITIES
   Capital expenditures:
      Acquisition of Tejas ...................................................   (1,376)             --
      Other ..................................................................   (1,188)         (1,485)
   Proceeds from property sales and salvage ..................................      196             127
   Other investments .........................................................      241            (202)
                                                                                -------        --------
              Net Cash Used for Investing Activities .........................   (2,127)         (1,560)
                                                                                -------        --------
CASH FLOWS PROVIDED BY (USED FOR) FINANCING ACTIVITIES
   Proceeds from issuance of long-term debt ..................................      324             161
   Principal payments on long-term debt ......................................     (370)           (467)
   Proceeds from sales of securities of subsidiaries .........................      246              32
   Dividends to shareholder ..................................................     (825)           (800)
   Dividends to minority interest ............................................      (46)            (31)
   Increase (decrease) in short-term obligations .............................    2,230           1,405
                                                                                -------        --------
              Net Cash Provided by Financing Activities ......................    1,559             300
                                                                                -------        --------
NET CASH FLOWS
   Increase (Decrease) in cash and cash equivalents .........................   $   (56)       $    257
                                                                                =======        ========
CASH AND CASH EQUIVALENTS
   Balance at beginning of period ............................................  $   342        $    393
   Increase (decrease) in cash and cash equivalents ..........................      (56)            257
                                                                                -------        --------
              Balance at end of period .......................................  $   286        $    650
                                                                                =======        ========
</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 second quarter and first six 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


         The following summary, prepared on a pro forma basis, presents the
Shell Oil results of operations for the three month and six month periods ending
June 30, 1997 as if Equilon 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 Month Period             Six Month Period
                             Ended June 30, 1997           Ended June 30, 1997
                             -------------------           -------------------
                                           (Millions of dollars)
<S>                              <C>                         <C>      
   Gross Revenues..........      $ 5,377                     $  11,840
   Net income..............          523                         1,033
</TABLE>


C.  EQUILON ENTERPRISES LLC

         The following unaudited financial information for Equilon is reflected
on a 100 percent Equilon basis:
                            
<TABLE>
<CAPTION>
                             Three Month Period             Six Month Period
                             Ended June 30, 1998           Ended June 30, 1998
                             -------------------           -------------------
                                           (Millions of dollars)
<S>                              <C>                       <C>       
   Gross Revenues..........      $6,070                    $   12,095
   Income Before Tax.......         198                           310
</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.


D.  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 


6


<PAGE>   7

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 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 reported second quarter net income of $316 million, a
decrease of $215 million, or 40 percent, from the second quarter of 1997.
Excluding special items in both quarters, adjusted net income in the second
quarter of 1998 totaled $312 million, a decrease of $164 million or 34 percent.

         The key operational elements contributing to lower earnings in the
second quarter of 1998 as compared to 1997 were lower average crude oil prices
and lower earnings from the chemical business. Domestic crude oil production
increased 13 percent over the same period last year, while natural gas
production rose 20 percent.

         For the first six months of 1998, net income was $488 million, a
decrease of $560 million, or 53 percent, from the same period last year.
Excluding special items, adjusted net income for 1998 totaled $483 million, a
decrease of $492 million, or 50 percent, from 1997.

         Contributing to the lower earnings in the first six months of 1998 were
lower average prices for crude oil and natural gas and lower earnings from the
chemical business. Results in oil products were 

7

<PAGE>   8


virtually unchanged during 1998 compared to 1997 due in part to a continuing
very competitive business environment.

         Special items in the 1998 periods benefited net income $4 million for
the quarter and $5 million for the first half. Special items in the second
quarter of 1998 were comprised primarily of gains from oil and gas property
sales offset in part by redundancy provisions. Special items increased 1997 net
income by $55 million for the quarter and $73 million in the first six months.


OIL AND GAS EXPLORATION AND PRODUCTION

<TABLE>
<CAPTION>
Income Highlights              Second Quarter                     Six Months
-----------------           --------------------            ---------------------
(millions of dollars)       1998            1997            1998             1997
                            ----            ----            ----             ----
<S>                          <C>             <C>             <C>              <C>
Segment Net Income .....     184             262             332              715
Special Items ..........      39              10              42               36
                             ---             ---             ---              ---
Adjusted Net Income ....     145             252             290              679
</TABLE>

         Oil and gas exploration and production net income in the second quarter
of 1998 totaled $184 million, a decrease of $78 million from 1997. For the first
half of 1998, earnings were $332 million, down $383 million. Excluding special
items in the comparable periods, adjusted net income declined $107 million in
the 1998 quarter versus 1997 and $389 million in the first-half comparison.

         Production of crude oil and natural gas in both 1998 periods increased
significantly; however, sharply lower crude oil prices more than offset these
gains. For the second quarter of 1998, domestic crude oil prices averaged $11.17
per barrel, decreasing $5.26 per barrel, or 32 percent, from the 1997 quarter.
For the first six months of 1998, average domestic crude oil prices decreased
$6.33 per barrel, or 35 percent. Operating expenses were higher in both 1998
periods, although on a per barrel equivalent basis, costs were lower.

         Average domestic crude oil production during the 1998 periods was
460,000 barrels per day for the quarter and 455,000 barrels per day for the
first six months, increasing 52,000 and 56,000 barrels per day, respectively,
compared to 1997. These increases resulted primarily from the deepwater Gulf of
Mexico development, and more than offset natural crude oil production declines
elsewhere. Natural gas production averaged 2,076 million cubic feet daily during
the second quarter of 1998, increasing 343 million cubic feet daily, or 20
percent. For the first six months of 1998, natural gas production averaged 1,987
million cubic feet daily, an increase of 241 million, or 14 percent. This year's
second quarter natural gas production represents an all-time high for Shell Oil.
These record levels are largely attributable to new and increased production
from deepwater Gulf of Mexico.

         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.

8

<PAGE>   9


DOWNSTREAM GAS

<TABLE>
<CAPTION>
Income Highlights                Second Quarter                 Six Months
-----------------            ----------------------         ------------------
(millions of dollars)        1998              1997         1998          1997
                             ----              ----         ----          ----
<S>                          <C>               <C>          <C>           <C>
Segment Net Income .......     14                --           23            --
Special Items ............     --                --           --            --
                             ----              ----         ----          ----
  Adjusted Net Income ....     14                --           23            --
</TABLE>

         Downstream gas, a new operating segment of Shell Oil, had earnings for
the second quarter of 1998 of $14 million, essentially unchanged from the first
quarter of this year. 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 second quarter of 1998, downstream gas transported natural
gas volumes were about 6.8 BCF/D, up slightly over the first quarter of 1998.
Gas transport margins were essentially unchanged from quarter to quarter;
however, gas processing margins declined.


OIL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                 Second Quarter                Six Months
-----------------              --------------------         ------------------
(millions of dollars)          1998            1997         1998          1997
                               ----            ----         ----          ----
<S>                             <C>             <C>          <C>           <C>
Segment Net Income .......      117             128          128           139
Special Items ............      (29)             (1)         (30)           (6)
                               ----            ----         ----          ----
  Adjusted Net Income ....      146             129          158           145
</TABLE>

         Oil products earnings, including Shell Oil's equity share of the
earnings of Equilon, totaled $117 million in the second quarter of 1998, a
decrease of $11 million from 1997. In the first six months of 1998, earnings
totaled $128 million, down $11 million from 1997. However, after excluding
special items in comparable periods, adjusted net income increased $17 million
versus the 1997 quarter, and $13 million in the six-month comparison. Special
items in 1998 included a severance provision associated with the new downstream
refining and marketing alliances.

         Overall, the key factor in the higher operating earnings in the 1998
periods compared to 1997 was an improvement in refined product margins. Earnings
in the second quarter of 1998 also improved over the prior quarter, mainly due
to higher margins and lower manufacturing costs associated with turnarounds.

         As further discussed in Note B of the Notes to Interim Financial
Statements and as reported in the Company's Current Report on Form 8-K filed
with the Commission on January 30, 1998, 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 

9

<PAGE>   10


United States refining and marketing businesses and their nationwide trading,
transportation and lubricants businesses.


CHEMICAL PRODUCTS

<TABLE>
<CAPTION>
Income Highlights                 Second Quarter            Six Months
-----------------              --------------------      ----------------
(millions of dollars)          1998            1997      1998        1997
                               ----            ----      ----        ----
<S>                            <C>             <C>       <C>         <C>
Segment Net Income .......       75             151       180         257
Special Items ............      (11)             (4)       (9)         (7)
                               ----            ----      ----        ----
  Adjusted Net Income ....       86             155       189         264
</TABLE>

         Chemical products earnings were $75 million in the second quarter of
1998, a decrease of $76 million from 1997. For the first six months of 1998,
chemical products earnings totaled $180 million, a decrease of $77 million.
Excluding special items in the comparable periods, adjusted net income for the
1998 quarter decreased $69 million and for the first six months decreased $75
million.

         Earnings declined in the second quarter of 1998 due to lower prices for
primary chemicals, which more than offset the benefits derived from lower
feedstock costs, and to a $9 million charge associated with the restructuring of
a business. For the six months of 1998, lower primary chemical margins and
higher operating expenses, primarily turnaround costs, contributed to the
earnings decline.


OTHER

         The other segment net income in the second quarter and first half of
1998 was $5 million and $9 million, respectively, compared to net losses of $2
million and $1 million in the in the same 1997 periods.


CORPORATE ITEMS

         Corporate items reduced earnings $79 million and $184 million in the
second quarter and first half of 1998, respectively, compared to a reduction to
earnings of $8 million and $62 million in the comparable 1997 periods. Excluding
special items, corporate charges totaled $84 million in the 1998 quarter and
$186 million in the first six months of 1998 compared to costs of $58 million
and $112 million in the same 1997 periods. Higher financing costs in both 1998
periods were the primary factor in the increases.


FINANCIAL CONDITION

CAPITAL RESOURCES AND LIQUIDITY

         Cash flow provided by operating activities totaled $512 million for the
first six months of 1998, compared with $1,517 million in the comparable period
last year, a decrease of $1,005 million. The period to period decrease was
attributable to lower earnings and, in part, to lower dividends from equity
investments. Cash generated from operating activities, coupled with an increase
in debt and sale of securities totaling $2,430 million, and proceeds from
property sales and other investments of $437 million in the first six months of
1998, was used primarily for capital expenditures of $2,564 million and dividend
payments of $871 million.

10

<PAGE>   11


OTHER MATTERS

RECENT DEVELOPMENTS

         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 U.S. 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 will account for its investment in Motiva using the 
equity method of accounting.

         On June 30, 1998 Shell Oil conveyed substantially all of its onshore
oil and gas property interests in south Louisiana (excluding offshore interests
and certain interests in Louisiana state waters and the onshore areas adjacent
to such waters) to The Meridian Resource Corporation (Meridian) in exchange for
cash, and common and preferred stock interests in Meridian as reported in the
Company's Schedule 13D filed with the Commission on July 10, 1998.


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.

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

11

<PAGE>   12


                           PART II. OTHER INFORMATION


ITEM 1.  LEGAL PROCEEDINGS.

         In July 1998, an exploration and production subsidiary of the Company
agreed to a civil penalty in the amount of $128 thousand, to be paid in cash or
by funding of a fugitive emission and odor study. The penalty was assessed by
the San Luis Obispo Air Pollution Control District.

         In 1998, the Shell Deer Park Chemical Plant was informed by the
Environmental Protection Agency ("EPA") that the agency is considering an
enforcement action under Section 112(r)(1) of the Clean Air Act Amendments of
1990. The EPA and Shell are engaging in discussions regarding such potential
alleged past violations.

ITEM 6.   EXHIBITS AND REPORTS ON FORM 8-K

         (a)    Exhibits.

                2.1      Asset Transfer and Liability Assumption Agreement
                         dated as of July 1, 1998 among Shell Oil, Texaco Inc.
                         and Saudi Arabian Oil Company for the creation of
                         Motiva Enterprises LLC.

                27.      Financial Data Schedule.

         (b)    Reports on Form 8-K.

                No Reports on Form 8-K were filed in the second quarter of 1998.
                However, on July 1, 1998, the Company filed a Current Report on
                Form 8-K regarding 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:  July 29, 1998

12


<PAGE>   13


                                 EXHIBIT INDEX
         
<TABLE>
<CAPTION>
EXHIBIT
NUMBER                                 DESCRIPTION
-------                                -----------
<S>                      <C>                                          
2.1                      Asset Transfer and Liability Assumption Agreement
                         dated as of July 1, 1998 among Shell Oil, Texaco Inc.
                         and Saudi Arabian Oil Company for the creation of
                         Motiva Enterprises LLC.

27.                      Financial Data Schedule.
</TABLE>