BARROW HANLEY MEWHINNEY & STRAUSS INC
SC 13G/A, 1999-02-12
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<PAGE>
 
                                  UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                  SCHEDULE 13G

                    Under the Securities Exchange Act of 1934
                               (Amendment No. 1)*

                               Triton Energy Ltd.
- --------------------------------------------------------------------------------
                                (Name of Issuer)


                         Common Stock (Ordinary Shares)
- --------------------------------------------------------------------------------
                         (Title of Class of Securities)


                                   G90751101
- --------------------------------------------------------------------------------
                                 (CUSIP Number)


                                December 31, 1998
- --------------------------------------------------------------------------------
             (Date of Event Which Requires Filing of this Statement)

Check appropriate box to designate the rule pursuant to which this Schedule is
filed:

         [x]  Rule 13d-1(b)

         [_]  Rule 13d-1(c)

         [_]  Rule 13d-1(d)

*The remainder of this cover page shall be filled out for a reporting person's
initial filing on this form with respect to the subject class of securities, and
for any subsequent amendment containing information which would alter
disclosures provided in a prior cover page.

The information required on the remainder of this cover page shall not be deemed
to be "filed" for the purpose of Section 18 of the Securities Exchange Act of
1934 ("Act") or otherwise subject to the liabilities of that section of the Act
but shall be subject to all other provisions of the Act (however, see the
Notes).

                               Page 1 of 4 Pages
<PAGE>
 
- -----------------------------
CUSIP No. G90751101                        13G
- -----------------------------



- -------------------------------------------------------------------------------
    1.    NAME OF REPORTING PERSON                                             
          S.S. OR I.R.S. IDENTIFICATION NO. OF ABOVE PERSON                    
                                                                               
            Barrow, Hanley, Mewhinney & Strauss, Inc.                          
                       752403190                                               
- -------------------------------------------------------------------------------
                                                        (a)     [_]
    2.    CHECK THE APPROPRIATE BOX IF A MEMBER OF A GROUP (See Instructions)

                                                        (b)     [_]
- -------------------------------------------------------------------------------
    3.    SEC USE ONLY

- -------------------------------------------------------------------------------
    4.    CITIZENSHIP OR PLACE OF ORGANIZATION

                   A Nevada corporation
- -------------------------------------------------------------------------------
                               5.    SOLE VOTING POWER
         NUMBER OF                            3,224,530 shares
          SHARES               ------------------------------------------------
       BENEFICIALLY            6.    SHARED VOTING POWER                       
         OWNED BY                             1,704,900 shares                 
           EACH                ------------------------------------------------
         REPORTING             7.    SOLE DISPOSITIVE POWER                    
          PERSON                              4,929,430 shares                 
           WITH                ------------------------------------------------
                               8.    SHARED DISPOSITIVE POWER                   
                                              ----                              
- ------------------------------------------------------------------------------- 
    9.   AGGREGATE AMOUNT BENEFICIALLY OWNED BY EACH REPORTING PERSON
               4,929,430 shares
- -------------------------------------------------------------------------------
   10.   CHECK IF THE AGGREGATE AMOUNT IN ROW (9) EXCLUDES
               CERTAIN SHARES (See Instructions)
- -------------------------------------------------------------------------------
   11.   PERCENT OF CLASS REPRESENTED BY AMOUNT IN ROW (9)      [_]
               13.5%
- -------------------------------------------------------------------------------
   12.   TYPE OF REPORTING PERSON (See Instructions)
               IA
- -------------------------------------------------------------------------------



                               Page 2 of 4 Pages
<PAGE>
 
SCHEDULE 13G
- ------------

Item 1(a)      Name of Issuer:
                       Triton Energy Ltd.

     1(b)      Address of Issuer's Principal Executive Offices:
                       6688 North Central Expressway
                       Suite 1400
                       Dallas, TX  75206

Item 2(a)      Name of Person Filing:
                       Barrow, Hanley, Mewhinney & Strauss, Inc.

     2(b)              Address of Principal Business Office or, if none,
                       Residence: One McKinney Plaza 3232 McKinney Avenue, 15th
                       Floor Dallas, TX 75204-2429

     2(c)      Citizenship:
                       A Nevada corporation

     2(d)      Title of Class of Securities:
                       Common Stock

     2(e)      CUSIP Number:
                       G90751101

Item 3         If this statement is filed pursuant to Rules 13d-1(b), or 13d-
               2(b):
                       The reporting person is an Investment Adviser registered
                       under Section 203 of the Investment Advisers Act of 1940.

Item 4         Ownership:

     4(a)      Amount beneficially owned:
                       4,929,430 shares

     4(b)      Percent of Class:
                       13.5%

     4(c)      Number of shares as to which such person has:

               (i)     Sole power to vote or to direct the vote:
                            3,224,530 shares


                               Page 3 of 4 Pages
<PAGE>
 
               (ii)    Shared power to vote or to direct the vote:
                              1,704,900 shares

               (iii)   Sole power to dispose or to direct the disposition of:
                              4,929,430 shares

               (iv)    Shared power to dispose or to direct the disposition of:
                              --

Item 5         Ownership of Five Percent or Less of a Class:
                              Not Applicable.

Item 6         Ownership of More than Five Percent on Behalf of Another Person:
                              The right to receive or the power to direct the
                              receipt of dividends from, or the proceeds from
                              the sale of, the common stock is held by certain
                              clients of the reporting person, none of which has
                              such right or power with respect to five percent
                              or more of the common stock.

Item 7         Identification and Classification of the Subsidiary which
               Acquired the Security Being Reported on by the Parent Holding
               Company:
                              Not Applicable.

Item 8         Identification and Classification of Members of the Group:
                              Not Applicable.

Item 9         Notice of Dissolution of Group:
                              Not Applicable.

Item 10        Certification:

               By signing below the undersigned certifies that, to the best of
               its knowledge and belief, the securities referred to above were
               acquired in the ordinary course of business and were not acquired
               for the purpose of and do not have the effect of changing or
               influencing the control of the issuer of such securities and were
               not acquired in connection with or as a participant in any
               transaction having such purpose or effect.

After reasonable inquiry and to the best of its knowledge and belief, the
undersigned certifies that the information set forth in this statement is true,
complete and correct.

                                 BARROW, HANLEY, MEWHINNEY &
                                 STRAUSS, INC.
                                 
                                 
                                 By: /s/ Bryant M. Hanley, Jr.
                                          Name: Bryant M. Hanley, Jr.
                                          Title:   President

February 11, 1999



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