<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, 1999.
                                       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, 1999 - 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           
                                                                           ---------------------------
                                                                            1999                 1998     
                                                                           -------             -------
<S>                                                                        <C>                 <C>    
REVENUES
   Sales and other operating revenue..................................     $ 3,488             $ 5,026
   Less:   Consumer excise and sales taxes............................          22                 378
                                                                           -------             -------
                                                                             3,466               4,648
   Equity in income of affiliates.....................................         139                 120
   Dividend, interest and other income................................           6                  36
                                                                           -------             -------
             TOTAL....................................................       3,611               4,804

COSTS AND EXPENSES
   Purchased raw materials and products...............................       2,077               2,945
   Operating expenses.................................................         649                 700
   Selling, general and administrative expenses.......................          87                 261
   Exploration, including exploratory dry holes.......................          35                  69
   Research expenses..................................................          32                  39
   Depreciation, depletion, amortization and retirements..............         290                 396
   Interest and discount amortization.................................          77                  85
                                                                           -------             -------
         TOTAL........................................................       3,247               4,495

INCOME BEFORE INCOME TAXES AND MINORITY INTEREST......................         364                 309

   Federal and other income taxes.....................................          24                 113
   Minority interest in income of subsidiaries........................          12                  24
                                                                           -------             -------

NET INCOME............................................................     $   328             $   172
                                                                           =======             =======
</TABLE>


Note: Certain 1998 amounts have been reclassified to conform with current year
presentation.



                                       2
<PAGE>   3


                       SHELL OIL COMPANY AND SUBSIDIARIES
                           CONSOLIDATED BALANCE SHEET
                               Millions of Dollars


<TABLE>
<CAPTION>
                                                             MARCH 31,   DECEMBER 31,
                                                              1999           1998 
                                                             ---------   ------------
<S>                                                          <C>         <C>         
ASSETS
CURRENT ASSETS
   Cash and cash equivalents .............................   $     310   $        322
   Receivables and prepayments, less allowance for
      doubtful accounts ..................................       2,274          2,235
   Owing by related parties ..............................         637            618
   Inventories of oils and chemicals .....................         740            723
   Inventories of materials and supplies .................         137            150
                                                             ---------   ------------
             TOTAL CURRENT ASSETS ........................       4,098          4,048
INVESTMENTS ..............................................       9,437          9,345
LONG-TERM RECEIVABLES AND DEFERRED CHARGES ...............       1,707          1,706
PROPERTY, PLANT AND EQUIPMENT AT COST, LESS
   ACCUMULATED DEPRECIATION, DEPLETION AND
   AMORTIZATION OF $12,865 AT MARCH 31, 1999
   AND $12,735 AT DECEMBER 31, 1998 ......................      11,591         11,444
                                                             ---------   ------------
             TOTAL .......................................   $  26,833   $     26,543
                                                             =========   ============


LIABILITIES AND SHAREHOLDER'S EQUITY
CURRENT LIABILITIES
   Accounts payable - trade ..............................   $   1,190   $      1,490
   Other payables and accruals ...........................       1,272          1,406
   Income, operating and consumer taxes ..................         178             88
   Owing to related parties ..............................         340            553
   Short-term debt .......................................       6,077          5,301
                                                             ---------   ------------
           TOTAL CURRENT LIABILITIES .....................       9,057          8,838
LONG-TERM DEBT ...........................................         486            490
DEFERRED INCOME TAXES ....................................       2,095          2,000
LONG-TERM LIABILITIES ....................................       2,372          2,287
MINORITY INTEREST AND PREFERRED STOCK OF SUBSIDIARIES ....       1,534          1,542
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 ...................................       9,083          9,180
                                                             ---------   ------------
           TOTAL SHAREHOLDER'S EQUITY ....................      11,289         11,386
                                                             ---------   ------------
           TOTAL .........................................   $  26,833   $     26,543
                                                             =========   ============
</TABLE>


                                       3
<PAGE>   4


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

<TABLE>
<CAPTION>
                                                                          FIRST QUARTER      
                                                                       ------------------
                                                                        1999        1998
                                                                       -------    -------
<S>                                                                    <C>        <C>    
CASH FLOWS FROM OPERATING ACTIVITIES
   Net Income ......................................................   $   328    $   172
   Adjustments to reconcile net income to net cash
       provided by operating activities:
          Depreciation, depletion, amortization and retirements ....       290        396
          Dividends in excess of (less than) equity income .........      (118)       (44)
          (Increases) decreases in working capital:
              Receivables and prepayments ..........................       (58)       425
              Inventories ..........................................        (4)       (43)
              Current payables and accruals ........................      (557)      (401)
          Deferred income taxes ....................................        95       (339)
          Minority interest in income of subsidiaries ..............        12         24
          Other noncurrent items ...................................        84        110
                                                                       -------    -------
              Net Cash Provided by Operating Activities ............        72        300

CASH FLOWS PROVIDED BY (USED FOR) INVESTING ACTIVITIES
   Capital Expenditures ............................................      (483)    (1,993)
   Proceeds from property sales and salvage ........................        51         67
   Other investments ...............................................        21        (69)
                                                                       -------    -------
              Net Cash Used for Investing Activities ...............      (411)    (1,995)

CASH FLOWS PROVIDED BY (USED FOR) FINANCING ACTIVITIES
   Principal payments on long-term debt ............................       (12)      (340)
   Proceeds from sales of securities of subsidiaries ...............        (4)       338
   Dividends to shareholder ........................................      (425)      (413)
   Dividends to minority interest ..................................       (16)       (27)
   Increase in short-term obligations ..............................       784      2,156
                                                                       -------    -------
              Net Cash Provided by Financing Activities ............       327      1,714

NET CASH FLOWS
   Increase (Decrease) in cash and cash equivalents ................   $   (12)   $    19
                                                                       =======    =======

CASH AND CASH EQUIVALENTS
   Balance at beginning of period ..................................   $   322    $   342
   Increase (decrease) in cash and cash equivalents ................       (12)        19
                                                                       -------    -------
              Balance at end of period .............................   $   310    $   361
                                                                       =======    =======
</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 in the Registrant's Annual Report on Form 10-K for the year ended
December 31, 1998. 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 1999 should not be construed as
necessarily indicative of future financial results.

B. EQUILON ENTERPRISES LLC

         Summarized unaudited financial information for Equilon Enterprises LLC
("Equilon"), formed January 1, 1998 and jointly owned 56 percent by Shell Oil
and 44 percent by Texaco Inc. ("Texaco") is presented below on a 100 percent
Equilon basis:

<TABLE>
<CAPTION>
                                                            Three Months Ended              Three Months Ended
                                                              March 31, 1999                  March 31, 1998
                                                              --------------                  --------------
<S>                                                         <C>                             <C>      
                                                                           (Millions of dollars)
      Gross Revenues...................................         $   5,779                        $   6,360
      Income Before Tax................................               171                              112
</TABLE>

         Shell Oil accounts for its interest in Equilon using the equity method
of accounting. Under this method, Shell Oil's share of Equilon's results of
operations is recorded on a one-line basis to "Equity in Income of Affiliates"
in the Consolidated Statement of Income. Additionally, since Equilon is a
limited liability company, Shell Oil records the provision for income taxes and
related liability applicable to its share of Equilon's income in its
consolidated financial statements.

C. MOTIVA ENTERPRISES LLC
         Summarized unaudited financial information for Motiva Enterprises LLC
("Motiva"), formed July 1, 1998 and jointly owned 35 percent by Shell Oil and
32.5 percent each by Texaco and by Saudi Refining, Inc., a corporate affiliate
of Saudi Aramco, is presented below on a 100 percent Motiva basis:

<TABLE>
<CAPTION>
                                                          Three Months Ended
                                                            March 31, 1999
                                                          ------------------
<S>                                                       <C>      
                                                         (Millions of dollars)
      Gross Revenues...................................       $   2,242
      Income Before Tax................................              41
</TABLE>

         Shell Oil accounts for its interest in Motiva using the equity method
of accounting. Under this method, Shell Oil's share of Motiva's results of
operations is recorded on a one-line basis to "Equity in Income of Affiliates"
in the Consolidated Statement of Income. Additionally, since Motiva is a limited
liability company, Shell Oil records the provision for income taxes and related
liability applicable to its share of Motiva's income in its consolidated
financial statements.


                                       5
<PAGE>   6


D. INTERIM SEGMENT INFORMATION

         The following segment information has been prepared in accordance with
Statement of Financial Accounting Standards ("SFAS") No. 131, "Disclosure about
Segments of an Enterprise and Related Information," which was adopted by Shell
Oil in 1998. The adoption of SFAS No. 131 did not result in a change in Shell
Oil's reportable segments; however, the previously reported net income for each
segment has been restated primarily to reflect the allocation to the segments of
interest income and interest expense previously reported as "Nonallocated
Items."

<TABLE>
<CAPTION>
                                                           Three months ended March 31, 1999
                                                           ---------------------------------
                                     Oil and     Downstream      Oil       Chemical      All       Nonalloc.
                                       Gas          Gas        Products    Products     Other        Items       Total
                                     --------    ----------    --------    --------    --------    ---------    --------
<S>                                  <C>         <C>           <C>         <C>         <C>         <C>          <C>     
                                                                      (millions of dollars)
Sales and other operating
  revenues, including intersegment   $    755    $    1,754    $    391    $  1,175    $    134    $      30    $  4,239
Intersegment sales and revenues          (342)          (56)        (63)       (213)        (84)         (15)       (773)
                                     --------    ----------    --------    --------    --------    ---------    --------
Total external revenues              $    413    $    1,698    $    328    $    962    $     50    $      15    $  3,466
                                     ========    ==========    ========    ========    ========    =========    ========

Segment net income (loss)            $    224    $       (7)   $     53    $     87    $     (3)   $     (26)   $    328
Segment assets                         10,080         4,403       5,806       4,783         340        1,421      26,833
</TABLE>


<TABLE>
<CAPTION>
                                                     Three months ended March 31, 1998 (restated)
                                                     --------------------------------------------
                                     Oil and     Downstream      Oil       Chemical      All       Nonalloc.
                                       Gas          Gas        Products    Products     Other        Items       Total
                                     --------    ----------    --------    --------    --------    ---------    --------
<S>                                  <C>         <C>           <C>         <C>         <C>         <C>          <C>     
                                                                      (millions of dollars)
Sales and other operating
  revenues, including intersegment   $    952    $    1,674    $  1,458    $  1,455    $    132    $      27    $  5,698
Intersegment sales and revenues          (377)          (28)       (140)       (377)       (116)         (12)     (1,050)
                                     --------    ----------    --------    --------    --------    ---------    --------
Total external revenues              $    575    $    1,646    $  1,318    $  1,078    $     16    $      15    $  4,648
                                     ========    ==========    ========    ========    ========    =========    ========

Segment net income (loss)            $    125    $       (5)   $     (1)   $     98    $      9    $     (54)   $    172
Segment assets                         12,390         4,492       7,837       5,402         323        1,528      31,972
</TABLE>


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


                                       6
<PAGE>   7


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 alleged 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 believes that the District Court decision was
incorrect and appealed the decision to the Third Circuit in September 1998.
Shell Oil also 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

         AS DISCUSSED IN NOTE D OF THE NOTES TO INTERIM FINANCIAL STATEMENTS,
1998 SEGMENT NET INCOME FOR SHELL OIL'S REPORTABLE OPERATING SEGMENTS HAS BEEN
RESTATED CONSISTENT WITH THE 1999 PRESENTATION REFLECTING THE ADOPTION OF
STATEMENT OF FINANCIAL ACCOUNTING STANDARDS NO. 131, `DISCLOSURES ABOUT SEGMENTS
OF AN ENTERPRISE AND RELATED INFORMATION."


         Shell Oil Company reported first quarter 1999 net income including
special items of $328 million, an increase of $156 million, or 91 percent, over
net income of $172 million in the same 1998 period. Excluding special items in
both quarters, adjusted net income in the first quarter of 1999 increased $23
million, or 13 percent from the comparable 1998 period.

         Earnings in the first quarter of 1999 increased over the same period
last year due largely to lower operating costs, primarily in the oil and gas
exploration and production operating segment, and higher margins for oil
products, primarily on the West Coast. Offsetting these improvements to a large
degree were lower domestic crude oil prices, which declined about 21 percent,
and natural gas prices, which fell 19 percent compared to the same period in
1998. Chemical product earnings continued to reflect the effects of lower
margins resulting from reduced selling prices.

         Special items in the first quarter of 1999 benefited net income $134
million compared to $1 million last year. Special items in the 1999 quarter
included a tax benefit relating to prior periods, favorable litigation
outcome and gains from asset sales.


OIL AND GAS EXPLORATION AND PRODUCTION

Income Highlights

<TABLE>
<CAPTION>
                                                                             First Quarter      
                                                                        -------------------------
                                                                         1999              1998 
                                                                        -------           -------
                                                                           (millions of dollars)
<S>                                                                     <C>               <C>    
     Segment Net Income...........................................      $   224           $   125
     Special Items................................................           95                 3
                                                                        -------           -------
     Adjusted Net Income..........................................          129               122
</TABLE>

         Oil and gas exploration and production earnings including special items
totaled $224 million for the first quarter 1999, an increase of $99 million over
the comparable 1998 quarter. Adjusted net income, which excludes special items,
was $129 million for the 1999 quarter, an increase of $7 million over the
comparable 1998 quarter.

         In the first quarter of 1999, benefits from lower operating costs more
than offset lower crude oil and natural gas prices and lower crude oil
production. Domestic crude oil prices in the 1999 quarter averaged $9.54 per
barrel, down $2.58 per barrel from the 1998 quarter. Natural gas prices were
also lower, averaging $1.71 per thousand cubic feet, as compared to $2.12 per
thousand cubic feet in the same 1998 period.

         Domestic crude oil production in the first quarter of 1999 averaged
431,000 barrels per day, a decrease of 20,000 barrels per day from 1998. The
lower production was primarily due to normal


                                       8
<PAGE>   9


declines, and to a lesser degree, field sales. Crude oil production in the
deepwater Gulf of Mexico was about the same period to period with production
from the Mars and Ram-Powell platforms offsetting declines elsewhere that
resulted primarily from downtime associated with facility improvements and
curtailments. Natural gas production increased 8 percent over the same period in
1998, averaging 2,045 million cubic feet per day. Increased deepwater gas
production, including increases from the Mensa platform, more than offset
natural production declines elsewhere.

         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. Equity companies are those companies in
which Shell Oil has significant influence but not control.


DOWNSTREAM GAS

Income Highlights

<TABLE>
<CAPTION>
                                                                 First Quarter      
                                                             ------------------------
                                                               1999             1998 
                                                             --------         -------
                                                               (millions of dollars)
<S>                                                          <C>              <C>     
     Segment Net Income..................................    $     (7)        $    (5)
     Special Items.......................................          --              --
                                                             --------         -------
     Adjusted Net Income.................................          (7)             (5)
</TABLE>

         Downstream Gas incurred a net loss of $7 million in the first quarter
of 1999, compared to a net loss of $5 million in last year's first quarter.

         Benefits derived from improved storage margins for natural gas and
lower operating costs and taxes, were more than offset by increased allocated
financing costs. Sales of natural gas increased in the first quarter of 1999
about 40 percent over the same period last year; however, these benefits were
offset by lower natural gas margins due to market demand being reduced by mild
weather. Gas processing margins in the 1999 period were unfavorably impacted by
low natural gas liquids prices, and volumes were off 3 percent.


OIL PRODUCTS

Income Highlights

<TABLE>
<CAPTION>
                                                                     First Quarter      
                                                                ------------------------
                                                                  1999             1998 
                                                                --------         -------
                                                                  (millions of dollars)
<S>                                                             <C>              <C>     
     Segment Net Income.......................................  $     53         $    (1)
     Special Items............................................        36              (1)
                                                                --------         -------
     Adjusted Net Income......................................        17              --
</TABLE>

         Shell Oil's oil products segment consists primarily of its share of
results from Equilon, which began operations in January, 1998 (Shell Oil
interest - 56 percent), Motiva, which began operations in July, 1998 (Shell Oil
interest - 35 percent), and the Deer Park Refining Limited Partnership (Shell
Oil interest - 50 percent). Shell Oil accounts for its investment in each of
these companies using the equity method of accounting.

         Oil products earnings totaled $53 million in the first quarter of 1999,
an increase of $54 million over the comparable 1998 period. Adjusted net income,
which excludes special items, was $17 million for the first quarter of 1999, up
$17 million.

         Costs were lower in the 1999 quarter as a result of synergies derived
from the downstream


                                       9
<PAGE>   10


alliances, while in the first quarter of 1998 costs were relatively high due to
transition and formation costs associated with the startup of Equilon. Refining
margins were higher, primarily on the U.S. West Coast during the latter part of
the quarter. Improved alliance-associated income was partly offset by a major
scheduled turnaround at the Deer Park Refinery.



CHEMICAL PRODUCTS

Income Highlights

<TABLE>
<CAPTION>
                                                                       First Quarter     
                                                                 ----------------------
                                                                  1999            1998  
                                                                 ------          ------
                                                                  (millions of dollars)
<S>                                                              <C>             <C>   
     Segment Net Income........................................  $   87          $   98
     Special Items ............................................       3               2
                                                                 ------          ------
     Adjusted Net Income.......................................      84              96
</TABLE>

         Chemical products earnings were $87 million in the first quarter of
1999 compared with $98 million in the same 1998 period. Adjusted net income,
which excludes special items, was $84 million for the 1999 quarter, down $12
million.

         Earnings declined quarter to quarter due to lower margins, as benefits
from lower feedstock costs were more than offset by reduced selling prices.
Lower operating costs, primarily related to fewer plant maintenance and
turnarounds relative to the 1998 quarter, offset the effects of decreased sales
volumes resulting from business structure changes.


OTHER

         The other operating segment incurred a loss of $3 million in the first
quarter of 1999, compared to net income of $9 million in 1998. The decline in
the period to period comparison was due to lower margins derived from service
operations.


NONALLOCATED ITEMS

         Nonallocated costs totaled $26 million in the first quarter of 1999,
lower by $28 million compared to the same period last year, primarily due to a
high level of costs in 1998 associated with the corporate brand initiative.



FINANCIAL CONDITION

CAPITAL RESOURCES AND LIQUIDITY

         Cash flow provided by operating activities totaled $72 million for the
first quarter of 1999, compared with $300 million last year, a decrease of $228
million. The decrease was due in part to increases in working capital and lower
selling prices. The major uses of cash generated from operating activities,
coupled with an increase in debt of $772 million and other proceeds, were for
capital expenditures of $483 million and dividend payments of $441 million.


                                       10
<PAGE>   11


OTHER MATTERS

RECENT DEVELOPMENTS

         On April 29, 1999, Shell Oil announced that it had entered into a
definitive agreement to sell to Apache Corporation a significant number of oil
and gas properties located on the outer continental shelf of the Gulf of Mexico.
The sales price for the properties is $715 million in cash plus one million
shares of Apache Corporation common stock. The transaction is expected to close
within 30 days. Included in the properties being sold are a portion of the oil
and gas properties which in 1998, were identified as "assets to be sold" and
which were written down to their estimated sales prices.


REPORT ON 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 has
independent responsibility for identification, analysis and remediation of Y2K
problems within such business, and in 1997, Year 2000 Teams were 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.
Additionally, 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 analyzed the use of
computer technology in the respective businesses to determine where there were
business operation issues associated with the Y2K problem. Priority in such
reviews was placed first on health, safety and environmental issues, then on
areas which could materially affect third parties or revenues. These reviews
necessarily included 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 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 analysis and inventory of potential problems has been
completed in all areas involving critical IT Systems and Imbedded Technology
Systems, with remediation requirements identified and remediation plans
developed. Remediation of Y2K problems in all such areas in which issues have
been identified is anticipated to be substantially completed by the end of June
1999, with the remaining remediation work expected to be completed during the
third quarter of 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 through the third quarter 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. Analyses and remediation of non-critical IT Systems, Imbedded
Technology Systems and Third Party interfaces will continue through the
remainder of the year.


                                       11
<PAGE>   12


         As to the Imbedded Technology Systems used in industrial automation
within Shell Oil, the various Shell Oil businesses have used several approaches
in the completion of the identification and analysis of the potential Y2K
problem in such systems, including field installation inspection, and inspection
of design blueprints, with field-testing to confirm the adequacy of design
information. Remediation of all material problems with this technology is
expected to be completed by July 1, 1999.

         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 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 and do not include
Y2K costs of equity companies. At March 31, 1999, approximately 80 percent of
such expenditure had been committed or incurred. Funding for these efforts has
been a part of the normal budgeting of current operations within each business.

         Beyond the internal programs in the Shell Oil businesses, significant
efforts are continuing in order 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. Physical testing of key interfaces and
ongoing discussions are continuing with certain Third Parties 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 insure that there is not a disruption in the flow of those products
or services that are essential to the ability of Shell Oil to carry on its
business operations. However, since Shell Oil will remain dependent on Third
Party confirmations in certain areas, appropriate contingency plans are being
developed in each of Shell Oil's businesses.

         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 completed during the first
quarter of 1999 of Shell Oil's major business units have further validated the
coverage of the Y2K remediation programs in meeting this objective. These audits
will continue during second quarter. As a contingency plan, steps are being
developed to manage this issue around key dates where unanticipated Y2K problems
may arise. Current 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 is continuing to review and develop
its current business operation contingency plans in order to understand what
additions may be appropriate to those plans to deal with residual risk
associated with Y2K problems. These plans 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


                                       12
<PAGE>   13


relationships with Third Parties, the objective of Shell Oil's Year 2000 Team is
the significant reduction or complete elimination of these risks. Shell Oil
expects to achieve this goal. As stated above, steps are being 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 E of
the Notes to Interim Financial Statements.

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


                                       13
<PAGE>   14


                           PART II. OTHER INFORMATION


ITEM 1.   LEGAL PROCEEDINGS.

None.

ITEM 6.   EXHIBITS AND REPORTS ON FORM 8-K

         (a)    Exhibits.

                27.  Financial Data Schedule.

         (b)    Reports on Form 8-K.
                None




                                   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, Chief Financial Officer and
                                      Controller (Principal Accounting and
                                      Duly Authorized Officer)


Date:  May 6, 1999


                                       14
<PAGE>   15


                                INDEX TO EXHIBITS


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