<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 March 31, 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 April
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>


                                                                      FIRST QUARTER
                                                                    -----------------
                                                                     1998      1997
                                                                    -------   -------
<S>                                                                 <C>       <C>    
REVENUES
     Sales and other operating revenue ..........................   $ 5,026   $ 8,393
     Less:  Consumer excise and sales taxes .....................       378       916
                                                                    -------   -------
                                                                      4,648     7,477
     Equity in income of affiliates .............................       120        96
     Interest and other income ..................................        36        59
                                                                    -------   -------
              TOTAL .............................................     4,804     7,632

COSTS AND EXPENSES
     Purchased raw materials and products .......................     2,943     5,128
     Operating expenses .........................................       678       734
     Selling, general and administrative expenses ...............       219       214
     Exploration, including exploratory dry holes ...............        69        78
     Research expenses ..........................................        37        36
     Depreciation, depletion, amortization and retirements ......       396       499
     Interest and discount amortization .........................        85        45
     Operating taxes ............................................        68       113
                                                                    -------   -------
              TOTAL .............................................     4,495     6,847

INCOME BEFORE INCOME TAXES AND MINORITY INTEREST ................       309       785

     Federal and other income taxes .............................       113       253
     Minority interest in income of subsidiaries ................        24        15
                                                                    -------   -------

NET INCOME ......................................................   $   172   $   517
                                                                    =======   =======
</TABLE>

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

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


                         OPERATING SEGMENTS INFORMATION
                               Millions of Dollars


<TABLE>
<CAPTION>

                                                                       FIRST QUARTER
                                                                    ------------------
                                                                      1998       1997
                                                                    -------    -------
<S>                                                                 <C>        <C>    
SEGMENT NET INCOME (LOSS)  (net of minority interest)
     Oil and Gas Exploration and Production .....................   $   147    $   453
     Downstream Gas .............................................         9         --
     Oil Products ...............................................        11         11
     Chemical Products ..........................................       105        106
     Other ......................................................         3          1
Corporate Items .................................................      (103)       (54)
                                                                    -------    -------

NET INCOME ......................................................   $   172    $   517
                                                                    =======    =======
</TABLE>


                                        2

<PAGE>   3






                       SHELL OIL COMPANY AND SUBSIDIARIES
                           CONSOLIDATED BALANCE SHEET
                               Millions of Dollars

<TABLE>
<CAPTION>


                                                                      MARCH 31     DECEMBER 31
                                                                        1998           1997
                                                                    ------------   ------------
<S>                                                                 <C>            <C>         
ASSETS

CURRENT ASSETS
     Cash and cash equivalents ..................................   $        361   $        342
     Receivables and prepayments, less allowance for
         doubtful accounts ......................................          2,699          3,414
     Owing by related parties ...................................            292            280
     Inventories of oils and chemicals ..........................            791            974
     Inventories of materials and supplies ......................            169            218
                                                                    ------------   ------------
              TOTAL CURRENT ASSETS ..............................          4,312          5,228
INVESTMENTS .....................................................         10,389          6,456
LONG-TERM RECEIVABLES AND
     DEFERRED CHARGES ...........................................          1,986          1,150
PROPERTY, PLANT AND EQUIPMENT AT COST, LESS
     ACCUMULATED DEPRECIATION, DEPLETION AND
     AMORTIZATION OF $13,946 AT MARCH 31, 1998
     AND $16,505 AT DECEMBER 31, 1997 ...........................         14,252         16,767
GOODWILL, NET ...................................................          1,033             --
                                                                    ------------   ------------
              TOTAL .............................................   $     31,972   $     29,601
                                                                    ============   ============



LIABILITIES AND SHAREHOLDER'S EQUITY

CURRENT LIABILITIES
     Accounts payable - trade ...................................   $      1,390   $      2,257
     Other payables and accruals ................................          1,352          1,281
     Income, operating and consumer taxes .......................            144            186
     Owing to related parties ...................................            390            303
     Short-term debt ............................................          6,381          3,539
                                                                    ------------   ------------
             TOTAL CURRENT LIABILITIES ..........................          9,657          7,566
LONG-TERM DEBT ..................................................            776            585
DEFERRED INCOME TAXES ...........................................          3,249          3,339
LONG-TERM LIABILITIES ...........................................          2,238          2,154
MINORITY INTEREST ...............................................          1,414          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 ........................................         12,432         12,672
                                                                    ------------   ------------
             TOTAL SHAREHOLDER'S EQUITY .........................         14,638         14,878
                                                                    ------------   ------------
             TOTAL ..............................................   $     31,972   $     29,601
                                                                    ============   ============
</TABLE>




                                        3

<PAGE>   4



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

<TABLE>
<CAPTION>


                                                                     FIRST QUARTER
                                                                  ------------------
                                                                     1998       1997
                                                                  -------    -------
<S>                                                               <C>        <C>    
CASH FLOWS FROM OPERATING ACTIVITIES
     Net Income ...............................................   $   172    $   517
     Adjustments to reconcile net income to net cash
          provided by operating activities:
          Depreciation, depletion, amortization and retirements       396        499
         Dividends in excess of (less than) equity income .....       (44)       (64)
         (Increases) decreases in working capital:
              Receivables and prepayments .....................       425        410
              Inventories .....................................       (43)      (268)
              Current payables and accruals ...................      (401)      (480)
         Deferred income taxes ................................      (339)        85
         Minority interest in income of subsidiaries ..........        24         15
         Other noncurrent items ...............................       110        (56)
                                                                  -------    -------
              Net Cash Provided by Operating Activities .......       300        658
CASH FLOWS PROVIDED BY (USED FOR) INVESTING ACTIVITIES
     Capital expenditures:
         Acquisition of Tejas .................................    (1,376)        --
         Other ................................................      (617)      (663)
     Proceeds from property sales and salvage .................        67         16
     Other investments ........................................       (69)         3
                                                                  -------    -------
              Net Cash Used for Investing Activities ..........    (1,995)      (644)
                                                                  -------    -------
CASH FLOWS PROVIDED BY (USED FOR) FINANCING ACTIVITIES
     Proceeds from issuance of long-term debt .................        --         31
     Principal payments on long-term debt .....................      (340)      (253)
     Proceeds from sales of securities of subsidiaries ........       338          1
     Dividends to shareholder .................................      (413)      (400)
     Dividends to minority interest ...........................       (27)       (13)
     Increase (decrease) in short-term obligations ............     2,156        866
                                                                  -------    -------
              Net Cash Provided by Financing Activities .......     1,714        232
                                                                  -------    -------
NET CASH FLOWS
     Increase (Decrease) in cash and cash equivalents .........   $    19    $   246
                                                                  =======    =======
CASH AND CASH EQUIVALENTS
     Balance at beginning of period ...........................   $   342    $   393
     Increase (decrease) in cash and cash equivalents .........        19        246
                                                                  -------    -------
              Balance at end of period ........................   $   361    $   639
                                                                  =======    =======
</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 first quarter of 1998 should not be construed as
necessarily indicative of future financial results.

B.  SUMMARIZED FINANCIAL INFORMATION - SHELL PIPE LINE CORPORATION

         The following summarized financial information for Shell Pipe Line
Corporation is presented here for the information of holders of Shell Pipe Line
Corporation's 7 1/2% Guaranteed Sinking Fund Debentures due 1999, which are
fully guaranteed by Shell Oil Company.


<TABLE>
<CAPTION>

                                                                           March 31     December 31
Millions of dollars                                                          1998           1997
                                                                         ------------   ------------

<S>                                                                      <C>            <C>         
Current assets .......................................................   $  135          $  111
Noncurrent assets ....................................................      774             777
Current liabilities ..................................................      164             179
Noncurrent liabilities ...............................................       76              77
</TABLE>



<TABLE>
<CAPTION>

                                                                                 First Quarter
                                                                         ---------------------------
Millions of dollars                                                          1998           1997
                                                                         ------------   ------------

<S>                                                                      <C>            <C>         
Revenue ..............................................................   $   94         $   95
Operating income .....................................................       47             49
Net income ...........................................................       37             36
</TABLE>



C.  SIGNIFICANT 1998 ALLIANCES AND ACQUISITIONS

         EQUILON ENTERPRISES LLC. On January 15, 1998, Shell Oil and Texaco Inc.
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.


                                        5

<PAGE>   6



         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.

         The following summary, prepared on a pro forma basis, presents the
Shell Oil results of operations for the first quarter of 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
                                                                               Ended March 31, 1997
                                                                               ---------------------
                                                                               (millions of dollars)

<S>                                                                            <C>     
         Gross Revenues......................................................        $  6,463
         Net Income..........................................................             510
</TABLE>

         The above pro forma financial information was prepared by Shell Oil
based on available information and upon certain assumptions which management
believes are reasonable. The pro forma financial information is presented for
informational purposes only and is not necessarily indicative of the operating
results that would have occurred had the transactions been consummated for the
period presented. Additionally, the pro forma financial information is not
intended to be a prediction of future results, and it does not reflect any
synergies that are expected to be achieved from the combined operations.

D.  EQUILON ENTERPRISES LLC

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

<TABLE>
<CAPTION>

                                                                                Three Month Period
                                                                               Ended March 31, 1998
                                                                               ---------------------
                                                                               (millions of dollars)

<S>                                                                            <C>      
         Gross Revenues......................................................       $   6,360
         Income Before Tax...................................................             112
</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.

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

                                        6

<PAGE>   7



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 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.


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



                                        7

<PAGE>   8





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


RESULTS OF OPERATIONS

         Shell Oil Company reported first quarter net income of $172 million, a
decrease of $345 million, or 67 percent, from the $517 million in the same 1997
period. Excluding special items in both quarters, adjusted net income in the
first quarter of 1998 declined $328 million, or 66 percent from the comparable
1997 period.

         Earnings in the first quarter of 1998 reflected a severe drop in crude
oil and natural gas prices compared to the same period last year. Crude oil
prices declined about 40 percent, while natural gas prices fell almost 30
percent. At the same time, margins for refined products were pressured by
rapidly falling selling prices and higher costs, resulting in lower than
expected downstream earnings. Among other factors contributing to the earnings
decline was higher interest costs due to increased debt associated with the
January 1998 acquisition of Tejas.


OIL AND GAS EXPLORATION AND PRODUCTION

<TABLE>
<CAPTION>

Income Highlights                                                         FIRST QUARTER
-----------------                                                      --------------------
                                                                         1998       1997
                                                                       --------   ---------
                                                                       (millions of dollars)

<S>                                                                    <C>         <C>    
Segment Net Income ...................................................   $   147   $   453
Special Items ........................................................         3        26
                                                                         -------   -------
Adjusted Net Income ..................................................       144       427
</TABLE>

         Oil and gas exploration and production net income totaled $147 million
for the first quarter 1998, a decrease of $306 million from the 1997 quarter.
Adjusted net income, which excludes special items, was $144 million for the 1998
quarter, down $283 million.

         In the first quarter of 1998, income benefited from increased
production of crude oil and natural gas. Domestic crude oil production in the
first quarter of 1998 increased 16 percent compared to the same period in 1997,
while natural gas production increased 8 percent. The increase in domestic crude
oil production represents the sixth consecutive quarterly increase and the
highest level for domestic oil production in the last 10 years. This strong
operating performance was, however, more than offset by the downward spiral of
crude oil prices which began in 1997 and continued throughout the first quarter
of 1998. Domestic crude oil prices in the 1998 quarter averaged $12.12 per
barrel, down $7.16 per barrel from the 1997 quarter. Natural gas prices were
also lower, averaging $2.12 per thousand cubic feet, as compared to $2.93 per
thousand cubic feet in the same 1997 period.

         Domestic crude oil production in the first quarter of 1998 averaged
451,000 barrels per day compared to 389,000 barrels per day in 1997, up 62,000
barrels per day, primarily from increased production in the deepwater Gulf of
Mexico. Natural gas production increased 142 million cubic feet per day, or 8
percent. Domestic crude oil and natural gas production include Shell Oil's net
production plus a prorata share, based on ownership interest, of domestic equity
companies' production. Equity companies are those companies in which Shell Oil
has significant influence but not control.





                                        8

<PAGE>   9



DOWNSTREAM GAS

<TABLE>
<CAPTION>

Income Highlights                                                          FIRST QUARTER
-----------------                                                      ---------------------
                                                                          1998      1997
                                                                       ---------   --------
                                                                       (millions of dollars)

<S>                                                                    <C>         <C>    
Segment Net Income ...................................................   $     9   $    --
Special Items ........................................................        --        --
                                                                         -------   -------
Adjusted Net Income ..................................................         9        --
</TABLE>

         Downstream gas, a new operating segment of Shell Oil, had net income of
$9 million for the first quarter of 1998. The downstream gas segment began
operations in January, 1998; as a result, no comparative earnings data are
available for 1997.

         In January 1998, Shell Oil acquired Tejas, including Tejas' interest in
Coral Energy, L.P. (Coral), as further discussed in Note C 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 first quarter of 1998, this new business segment had pipeline
throughput of 6.6 BCF/D. Gas processing margins in the 1998 period were
unfavorably impacted by general market conditions. Gas transport margins were
also lower, due in part to the expiration of several long-term contracts.


OIL PRODUCTS

<TABLE>
<CAPTION>

Income Highlights                                                           FIRST QUARTER
-----------------                                                        -------------------
                                                                           1998      1997
                                                                         -------    -------
                                                                        (millions of dollars)

<S>                                                                      <C>        <C>    
Segment Net Income ...................................................   $    11    $    11
Special Items ........................................................        (1)        (5)
                                                                         -------    -------
Adjusted Net Income ..................................................        12         16
</TABLE>

         Oil products net income, including Shell Oil's equity share of the
earnings of Equilon totaled $11 million in the first quarter of 1998, the same
as in 1997. Adjusted net income, which excludes special items, was $12 million
for the first quarter of 1998, down $4 million.

         Overall, benefits derived from slightly improved margins were offset by
increased manufacturing costs, higher promotional expenses and transition costs.
Equilon's results benefited from strong performances by its transportation and
lubricants businesses.

         As further discussed in Note C 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 United States refining and marketing businesses and their nationwide
trading, transportation and lubricants businesses.









                                        9

<PAGE>   10




CHEMICAL PRODUCTS

<TABLE>
<CAPTION>

Income Highlights                                                                    FIRST QUARTER
-----------------                                                               -------------------------
                                                                                  1998             1997
                                                                                ---------       ---------
                                                                                  (millions of dollars)

<S>                                                                              <C>             <C>     
     Segment Net Income......................................................... $   105         $    106
     Special Items .............................................................       2               (3)
                                                                                 -------         ---------
     Adjusted Net Income........................................................     103              109
</TABLE>

         Chemical products net income was $105 million in the first quarter of
1998 compared with $106 million in the same 1997 period. Adjusted net income,
which excludes special items, was $103 million for the 1998 quarter, down $6
million.

         Downstream chemicals (intermediates, solvents and polymers) earnings
improved as margins benefited from lower feedstock costs. While primary
chemicals also benefited from lower feedstock costs, margins declined as selling
prices dropped more than feedstocks, particularly in Far East markets. Overall
margins improved, but these benefits were offset by lower sales volumes and
increased manufacturing turnaround costs.

         The olefins unit at the Deer Park, Texas plant, which was damaged in a
June 1997 explosion and fire, resumed operations during the quarter.


OTHER

         Net income for the other operating segment was $3 million in the first
quarter of 1998 compared to net income of $1 million in the first quarter of
1997.


CORPORATE ITEMS

         Corporate charges totaled $103 million in the first quarter of 1997
compared to charges of $54 million in the first quarter of 1997. Factors
contributing to these increased costs were higher financing costs related to a
higher level of debt and costs associated with the corporate brand initiative.


FINANCIAL CONDITION

CAPITAL RESOURCES AND LIQUIDITY

         Cash flow provided by operating activities totaled $300 million for the
first quarter of 1998, compared with $658 million last year, a decrease of $358
million. The decrease was due in part to lower earnings in the 1998 first
quarter. The major uses of cash generated from operating activities, coupled
with an increase in debt of $1,816 million and other proceeds, were for capital
expenditures of $1,993 million and dividend payments of $440 million.


OTHER MATTERS

RECENT DEVELOPMENTS

         During the second quarter of 1998, Shell Oil entered into an agreement
to sell its Anacortes refinery to Tesoro Petroleum Corporation. Shell Oil is
required to divest the refinery as part of a settlement with the Federal Trade
Commission ("FTC") and the attorneys general of the states of Washington and
Oregon involving Shell Oil's and Texaco's Equilon joint venture. Shell Oil and
Texaco have agreed that any gain or loss which may result from the sale of the
refinery will be deemed to be incurred by Equilon. Tesoro will acquire the
refinery for $237 million plus an additional payment for

                                       10

<PAGE>   11



net working capital. The FTC and the states will have final approval of the
transaction and Tesoro as the buyer. The transaction is expected to close during
the summer of 1998.

         Shell Oil, Texaco and Saudi Refining Incorporated (a corporate
affiliate of Saudi Aramco) are working to finalize agreements for a separate
joint venture involving their eastern United States and Gulf Coast refining and
marketing businesses. This venture is planned to be owned 35 percent by Shell
Oil, 32.5 percent by Texaco and 32.5 percent by Saudi Refining, Inc. (such
ownership to be subject to adjustment in the future based on the performance of
the assets).

         The Shell Oil Board of Directors has approved entering into equity swap
agreements based on Royal Dutch Petroleum Company stock. These swaps would not
exceed the amount of Royal Dutch Petroleum Company stock appreciation rights
granted to certain Shell Oil employees, currently approximately 9 million
shares, or exceed a five-year term. Actual execution of these swaps will be
dependent upon future market conditions and other factors.

         Net Cash Provided by Operating Activities was incorrectly reported in
Shell Oil's April 22, 1998 press release reporting first quarter 1998 earnings,
and has been corrected in this filing.

         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
international governmental entities, including those matters discussed in Note E
of the Notes to Interim Financial Statements. 

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



                           PART II. OTHER INFORMATION

ITEM 1.   LEGAL PROCEEDINGS

         As previously reported, the Shell Wood River Refining Company ("Wood
River") had received notice from the Office of the United States Attorney,
Southern District of Illinois, informing of that office's consideration of
filing of federal charges against Wood River alleging violations of the
migratory Bird Treaty Act. Wood River, the U.S. Attorney's Office, and the U. S.
Fish and Wildlife Service, on whose behalf the U.S. Attorney was acting, have
now resolved this matter. In response to violation notices issued by the Fish
and Wildlife Service, Wood River has forfeited collateral in the amount of
$20,000 to achieve this resolution.


ITEM 6.   EXHIBITS AND REPORTS ON FORM 8-K

         (a)    Exhibits.

                27. Financial Data Schedule.

         (b) Reports on Form 8-K.

                On January 30, 1998, the Company filed a Current Report on Form
         8-K for the following event:

         Item 2. Acquisition or Disposition of Assets and Item 7. Financial
         Statements and Exhibits regarding its January 15, 1998 acquisition and
         disposition of assets incidental to the formation of Equilon
         Enterprises LLC with Texaco Inc. The Form 8-K included unaudited pro
         forma consolidated financial statements of the Company for the year
         ended December 31, 1996 and the nine months ended September 30, 1997
         and were presented as if the formation and

                                       11

<PAGE>   12



         operational startup had occurred at the beginning of each period
         presented. As an exhibit to the 8-K the Company included the Asset
         Transfer and Liability Assumption Agreement dated as of January 15,
         1998 setting forth the terms of such acquisition and disposition.




                                   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:  May 8, 1998

                                       12

<PAGE>   13


                                INDEX TO EXHIBITS

<TABLE>
<CAPTION>

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



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