<PAGE>   1
                                 UNITED STATES
                       SECURITIES AND EXCHANGE COMMISSION
                            Washington, D.C.  20549

                                   Form 10-Q

             [X]  Quarterly Report Pursuant to Section 13 or 15(d)
                     of the Securities Exchange Act of 1934
                  For the quarterly period ended June 30, 1998

                                       or

             [ ]  Transition Report Pursuant to Section 13 or 15(d)
                     of the Securities Exchange Act of 1934
                    For the transition period from        to

                        Commission File Number: 1-12058

                BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST
             (Exact name of registrant as specified in its charter)


         Delaware                                         76-6088828
(State or other jurisdiction                           (I.R.S. Employer
    of incorporation or                                Identification No.)
       organization)

                               NationsBank, N.A.
                               NationsBank Plaza
                          901 Main Street, Suite 1700
                              Dallas, Texas  75202
                    (Address of principal executive offices)
                                   (Zip code)

                                 (214) 508-2304
              (Registrant's telephone number, including area code)

       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 [ ]

       Number of units of beneficial interest outstanding at August 1, 1998:
8,800,000
<PAGE>   2
                         PART I - FINANCIAL INFORMATION


Item 1.       Financial Statements.

       The condensed financial statements included herein have been prepared by
NationsBank, N.A. (as successor to NationsBank of Texas, N.A.), as Trustee (the
"Trustee") of Burlington Resources Coal Seam Gas Royalty Trust (the "Trust"),
pursuant to the rules and regulations of the Securities and Exchange
Commission.  Certain information and footnote disclosures normally included in
annual financial statements have been condensed or omitted pursuant to such
rules and regulations, although the Trustee believes that the disclosures are
adequate to make the information presented not misleading.  The condensed
financial statements of the Trust presented herein are unaudited, except for
balances as of December 31, 1997, and therefore are subject to year-end
adjustments.  It is suggested that these condensed financial statements be read
in conjunction with the financial statements and notes thereto in the Trust's
Annual Report on Form 10-K for the year ended December 31, 1997.  The December
31, 1997 balance sheet is derived from the audited balance sheet of that date. 
In the opinion of the Trustee, all adjustments necessary to present fairly the
assets, liabilities and trust corpus of the Trust as of June 30, 1998, the
distributable income for the three month and six month periods ended June 30,
1998 and 1997 and the changes in trust corpus for the six month periods ended
June 30, 1998 and 1997, have been included.  The distributable income for such
interim periods is not necessarily indicative of the distributable income for
the full year.

       The condensed financial statements as of June 30, 1998 and for the three
month and six month periods ended June 30, 1998 and 1997, included herein, have
been reviewed by Deloitte & Touche LLP, independent public accountants, as
stated in their report appearing herein.





                                       2
<PAGE>   3
                        Independent Accountants' Report


NationsBank, N.A.,
 as Trustee of Burlington Resources
 Coal Seam Gas Royalty Trust

We have reviewed the accompanying condensed statement of assets, liabilities
and trust corpus of the Burlington Resources Coal Seam Gas Royalty Trust as of
June 30, 1998, and the related condensed statements of distributable income for
the three month and six month periods ended June 30, 1998 and 1997 and changes
in trust corpus for the six month periods ended June 30, 1998 and 1997.  These
condensed financial statements are the responsibility of the Trustee.

We conducted our review in accordance with standards established by the
American Institute of Certified Public Accountants.  A review of interim
financial information consists principally of applying analytical procedures to
financial data and of making inquiries of persons responsible for financial and
accounting matters.  It is substantially less in scope than an audit conducted
in accordance with generally accepted auditing standards, the objective of
which is the expression of an opinion regarding the financial statements taken
as a whole.  Accordingly, we do not express such an opinion.

As described in Note 2 to the condensed financial statements, these condensed
financial statements have been prepared on a modified cash basis of accounting,
which is a comprehensive basis of accounting other than generally accepted
accounting principles.

Based on our review, we are not aware of any material modifications that should
be made to such condensed financial statements for them to be in conformity
with the basis of accounting described in Note 2.

We have previously audited, in accordance with generally accepted auditing
standards, the statement of assets, liabilities and trust corpus of Burlington
Resources Coal Seam Gas Royalty Trust as of December 31, 1997, and the related
statements of distributable income and changes in trust corpus for the year
then ended (not presented herein); and in our report dated March 27, 1998, we
expressed an unqualified opinion on those financial statements.  In our
opinion, the information set forth in the accompanying condensed statement of
assets, liabilities and trust corpus as of December 31, 1997, is fairly stated,
in all material respects, in relation to the statement of assets, liabilities
and trust corpus from which it has been derived.

/s/ Deloitte & Touche LLP
    Dallas, Texas


August 10, 1998





                                       3
<PAGE>   4
BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST

CONDENSED STATEMENTS OF ASSETS, LIABILITIES AND TRUST CORPUS



<TABLE>
<CAPTION>
                                                     June 30,        December 31,
                                                      1998              1997 
                                                   -----------       -----------
                                                   (unaudited)
<S>                                                <C>               <C>        
ASSETS

Cash and cash equivalents                          $    16,654       $    50,819
Royalty interests in gas
       properties (less accumulated
       amortization of $96,090,795
       at June 30, 1998 and $86,861,742
       at December 31, 1997)                        84,309,205        93,538,258
                                                   -----------       -----------

TOTAL ASSETS                                       $84,325,859       $93,589,077
                                                   ===========       ===========



LIABILITIES AND TRUST CORPUS

Trust expenses payable                             $   126,505       $   104,149

Trust corpus -
       8,800,000 units of beneficial
       interest authorized, issued and
       outstanding                                  84,199,354        93,484,928
                                                   -----------       -----------

TOTAL LIABILITIES
       AND TRUST CORPUS                            $84,325,859       $93,589,077
                                                   ===========       ===========
</TABLE>



              The accompanying notes are an integral part of these
                        condensed financial statements.





                                       4
<PAGE>   5
BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST

CONDENSED STATEMENTS OF DISTRIBUTABLE INCOME (UNAUDITED)


<TABLE>
<CAPTION>
                                                   FOR                  FOR
                                                THE THREE            THE THREE
                                               MONTHS ENDED         MONTHS ENDED
                                               June 30, 1998       June 30, 1997 
                                               -------------       ------------- 
<S>                                             <C>                 <C>        
Royalty income                                  $ 2,028,502         $ 1,618,799
Interest income                                       5,750               3,912
                                                -----------         -----------
                                                  2,034,252           1,622,711

General and administrative
       expenses                                    (222,080)           (222,234)
                                                -----------         -----------
Distributable income                            $ 1,812,172         $ 1,400,477
                                                ===========         ===========

Distributable income per
       unit (8,800,000 units)                   $       .21         $       .16
                                                ===========         ===========
</TABLE>


              The accompanying notes are an integral part of these
                        condensed financial statements.




                                       5
<PAGE>   6
BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST

CONDENSED STATEMENTS OF DISTRIBUTABLE INCOME (UNAUDITED)



<TABLE>
<CAPTION>
                                                    FOR                 FOR
                                                  THE SIX             THE SIX
                                                MONTHS ENDED        MONTHS ENDED
                                               June 30, 1998       June 30, 1997 
                                               -------------       ------------- 
<S>                                             <C>                 <C>        
Royalty income                                  $ 3,452,856         $ 3,090,719
Interest income                                      10,199               7,589
                                                -----------         -----------
                                                  3,463,055           3,098,308

General and administrative
       expenses                                    (408,273)           (418,520)
                                                -----------         -----------
Distributable income                            $ 3,054,782         $ 2,679,788
                                                ===========         ===========

Distributable income per
       unit (8,800,000 units)                   $       .35         $       .30
                                                ===========         ===========
</TABLE>




              The accompanying notes are an integral part of these
                        condensed financial statements.





                                       6
<PAGE>   7
BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST

CONDENSED STATEMENTS OF CHANGES IN TRUST CORPUS (UNAUDITED)



<TABLE>
<CAPTION>
                                                   FOR                 FOR
                                                 THE SIX             THE SIX
                                               MONTHS ENDED        MONTHS ENDED
                                              June 30, 1998       June 30, 1997 
                                              -------------       -------------
<S>                                           <C>                 <C>          
Trust corpus, beginning
       of period                              $  93,484,928       $ 107,328,165
Amortization of royalty
       interest                                  (9,229,053)         (8,030,508)
Distributable income                              3,054,781           2,679,788
Distributions to
       unitholders                               (3,111,302)         (2,715,387)
                                              -------------       -------------
Trust corpus, end
       of period                              $  84,199,354       $  99,262,058
                                              =============       =============
Distributions per unit
       (8,800,000 units)                      $         .35       $         .31
                                              =============       =============
</TABLE>



              The accompanying notes are an integral part of these
                         condensed financial statements.





                                       7
<PAGE>   8
BURLINGTON RESOURCES COAL SEAM GAS ROYALTY TRUST

NOTES TO CONDENSED FINANCIAL STATEMENTS (UNAUDITED)

1.     TRUST ORGANIZATION AND PROVISIONS

       Burlington Resources Coal Seam Gas Royalty Trust (the "Trust") was
formed as a Delaware business trust pursuant to the terms of the Trust
Agreement of Burlington Resources Coal Seam Gas Royalty Trust (the "Trust
Agreement") entered into effective as of May 1, 1993 by and among Burlington
Resources Oil & Gas Company, a Delaware corporation ("BROG"), as trustor,
Burlington Resources Inc., a Delaware corporation ("Burlington Resources"), and
NationsBank, N.A. (as successor to NationsBank of Texas, N.A.), a national
banking association (the "Trustee"), and Mellon Bank (DE) National Association,
a national banking association (the "Delaware Trustee"), as trustees.  The
trustees are independent financial institutions. Effective January 1, 1996,
Meridian Oil Production Inc. ("MOPI") was merged with and into Meridian Oil
Inc. ("MOI"), a wholly-owned subsidiary of Burlington Resources.  Effective
July 11, 1996, MOI changed its name to Burlington Resources Oil & Gas Company
and Meridian Oil Trading Inc. ("MOTI") and Meridian Oil Gathering Inc.
("MOGI"), both affiliates of MOI, changed their names to Burlington Resources
Trading Inc. ("BRTI") and Burlington Resources Gathering Inc. ("BRGI"),
respectively.  Accordingly, with regard to the period prior to the date of such
name changes, references in this Form 10-Q to BROG refer to MOPI, references to
BRTI refer to MOTI and references to BRGI refer to MOGI.

       The Trust is a grantor trust formed to acquire and hold certain net
profits interests (the "Royalty Interests") in BROG's interest in the Fruitland
coal formation underlying the Northeast Blanco Unit in the San Juan Basin of
New Mexico (the "Underlying Properties").  The Trust was initially created by
the filing of a Certificate of Trust with the Secretary of State of Delaware on
May 5, 1993.  In accordance with the Trust Agreement, BROG contributed $1,000
as the initial trust corpus of the Trust.  On June 17, 1993, the Royalty
Interests were conveyed to the Trust by BROG pursuant to the Net Profits
Interest Conveyance (the "Conveyance") dated effective as of May 1, 1993, in
consideration for all 8,800,000 authorized units of beneficial interest
("Units") in the Trust.  BROG transferred its Units by dividend to its parent,
Burlington Resources, Inc., which sold such Units to the public at $20.50 per
Unit through various underwriters in June 1993 (the "Public Offering") .  All
of the production attributable to the Underlying Properties is from the
Fruitland coal formation and currently constitutes "coal seam" gas that
entitles the owners of such production, provided certain requirements are met,
to tax credits pursuant to Section 29 of the Internal Revenue Code of 1986, as
amended.

       Royalty income to the Trust is attributable to the sale of depleting
assets.  All of the Underlying Properties burdened by the NPI (as hereinafter
defined) consist of producing properties.  Accordingly, the proved reserves
attributable to BROG's interest in the Underlying Properties are expected to
decline substantially during the term of the Trust and a portion of each cash
distribution made by the Trust will, therefore, be analogous to a return of
capital.  Accordingly, cash yields attributable to the Units are expected to
decline over the term of the Trust.





                                       8
<PAGE>   9
       The Trustee has all powers to collect and distribute proceeds received
by the Trust and to pay Trust liabilities and expenses.  The Delaware Trustee
has only such powers as are set forth in the Trust Agreement or are required by
law and is not empowered to otherwise manage or take part in the business of
the Trust.  The Royalty Interests are passive in nature and neither the
Delaware Trustee nor the Trustee has any control over or any responsibility
relating to the operation of the Underlying Properties or BROG's interest
therein.

       The Trust will terminate no later than December 31, 2012, subject to
earlier termination under certain circumstances described in the Trust
Agreement (the "Termination Date"). Cancellation of the Trust will occur on or
following the Termination Date when all Trust assets have been sold and the net
proceeds thereof are distributed to the holders of the Units ("Unitholders").

       The only assets of the Trust, other than cash and cash equivalents being
held for the payment of expenses and liabilities and for distribution to
Unitholders, are the Royalty Interests.  The Royalty Interests consist
primarily of a net profits interest (the "NPI") in BROG's interest in the
Underlying Properties.  The NPI generally entitles the Trust to receive 95
percent of the NPI Net Proceeds, as defined below.  The Royalty Interests also
include a 20 percent interest in the Infill Net Proceeds, as defined below,
from the sale of production if well spacing rules are effectively modified and
additional wells are drilled on producing drilling blocks in the Northeast
Blanco Unit ("Infill Wells") during the term of the Trust.  With respect to the
NPI, the term "NPI Net Proceeds" generally means the aggregate proceeds
attributable to BROG's net revenue interest in the Underlying Properties
(excluding the proceeds, if any, from Infill Wells) calculated at the price
paid by BRTI at any one of four central delivery points in the Northeast Blanco
Unit gathering system or either of two wellhead delivery points (collectively,
the "Central Gathering Point") for the entitled volume of gas produced and sold
from BROG's interest in the Underlying Properties less BROG's working interest
share of (i) property, production and related taxes (including severance
taxes); (ii) lease operating expenses; (iii) capital costs (if paid after
January 1, 1994); (iv) royalties, if any, required to be paid that are based on
the value of Section 29 tax credits attributable to such working interest
share; and (v) interest on the unrecovered portion, if any, of the foregoing
costs at a rate equal to the base rate (compounded quarterly) as announced from
time to time by Citibank, N.A. ("Citibank's Base Rate").  The term "Infill Net
Proceeds" generally means the aggregate proceeds attributable to BROG's net
revenue interest calculated at the price paid by BRTI at the Central Gathering
Point for the entitled volume of gas produced and sold from BROG's interest in
any Infill Wells less BROG's working interest share of (a) property, production
and related taxes (including severance taxes) on such Infill Wells; (b) lease
operating expenses with respect to such Infill Wells; (c) capital costs with
respect to such Infill Wells; and (d) interest on the unrecovered portion, if
any, of the foregoing costs at Citibank's Base Rate.  The complete definitions
of NPI Net Proceeds and Infill Net Proceeds are set forth in the Conveyance.
The definitions, formulas and accounting procedures and other terms governing
the computation of the Royalty Interests are set forth in the Conveyance.

       Because of the passive nature of the Trust and the restrictions and
limitations on the powers and activities of the Trustee contained in the Trust
Agreement, the Trustee does not consider any of the officers and employees of
the Trustee to be "officers" or "executive officers"





                                       9
<PAGE>   10
of the Trust as such terms are defined under the applicable rules and
regulations adopted under the Securities Exchange Act of 1934.

2.     BASIS OF ACCOUNTING

       The financial statements of the Trust are prepared on a modified cash
basis and are not intended to present financial position and results of
operations in conformity with generally accepted accounting principles
("GAAP").  Preparation of the Trust's financial statements on such basis
includes the following:

-      Royalty income and interest income are recorded in the period in which
       amounts are received by the Trust rather than in the months of
       production.

-      General and administrative expenses recorded are based on liabilities
       paid and cash reserves established out of cash received.

-      Amortization of the Royalty Interests is calculated on a unit-of-
       production basis and charged directly to trust corpus when revenues are
       received.

-      Distributions to Unitholders are recorded when declared by the Trustee
       (see Note 4).

       The financial statements of the Trust differ from financial statements
prepared in accordance with GAAP because royalty income is not accrued in the
period of production, general and administrative expenses recorded are based on
liabilities paid and cash reserves established rather than on an accrual basis,
and amortization of the Royalty Interests is not charged against operating
results.

       The net amount of royalty interests in gas properties is limited to the
sum of the future net cash flows attributable to the Trust's gas reserves at
year end using current unescalated product prices plus the estimated future
Section 29 credits for federal income tax purposes.  If the net cost of royalty
interests in gas properties exceeds this amount, an impairment provision will
be recorded and charged to the trust corpus.





                                       10
<PAGE>   11
3.     FEDERAL INCOME TAXES

       The Trust is a grantor trust for Federal income tax purposes.  As a
grantor trust, the Trust will not be required to pay Federal or state income
taxes.  Accordingly, no provision for income taxes has been made in these
condensed financial statements.

       Because the Trust will be treated as a grantor trust, and because a
Unitholder will be treated as directly owning an interest in the Royalty
Interests, each Unitholder will be taxed directly on his per Unit share of
income attributable to the Royalty Interests consistent with the Unitholder's
method of accounting and without regard to the taxable year or accounting
method employed by the Trust.

       Production from coal seam gas wells drilled after December 31, 1979 and
prior to January 1, 1993, qualifies for the Federal income tax credit for
producing nonconventional fuels under Section 29 of the Internal Revenue Code.
This tax credit is calculated annually based on each year's qualified
production through the year 2002.  Such credit, based on the Unitholder's pro
rata share of qualifying production, may not reduce his regular tax liability
(after the foreign tax credit and certain other non-refundable credits) below
his alternative minimum tax.  Any part of the Section 29 credit not allowed for
the tax year solely because of this limitation is subject to certain carryover
provisions.  Each Unitholder should consult his tax advisor regarding Trust tax
compliance matters.

4.     DISTRIBUTIONS TO UNITHOLDERS

       The Trustee determines for each quarter the amount of cash available for
distribution to Unitholders.  Such amount (the "Quarterly Distribution Amount")
is an amount equal to the excess, if any, of the cash received by the Trust, on
or before the last business day before the 50th day following the end of each
calendar quarter from the Royalty Interests attributable to production during
such quarter, plus, with certain exceptions, any other cash receipts of the
Trust during such quarter, over the liabilities of the Trust paid during such
quarter, subject to adjustments for changes made by the Trustee during such
quarter in any cash reserves established for the payment of contingent or
future obligations of the Trust.

       The Quarterly Distribution Amount for each quarter is payable to
Unitholders of record





                                       11
<PAGE>   12
on the 63rd day following the end of such calendar quarter unless such day is
not a business day in which case the record date is the next business day
thereafter.  The Trustee distributes the Quarterly Distribution Amount on or
prior to the 75th day after the end of each calendar quarter to each person who
was a Unitholder of record on the associated record date, together with
interest estimated to be earned on such amount from the date of receipt thereof
by the Trustee to the payment date.

       The Royalty Interests may be sold under certain circumstances and will
be sold following termination of the Trust.  A special distribution will be
made of undistributed net sales proceeds and other amounts received by the
Trust aggregating in excess of $10,000,000 (a "Special Distribution Amount").
The record date for a Special Distribution Amount will be the 15th day
following receipt of amounts aggregating a Special Distribution Amount by the
Trust (unless such day is not a business day in which case the record date will
be the next business day thereafter) unless such day is within 10 days prior to
the record date for a Quarterly Distribution Amount in which case the record
date will be the date as is established for the next Quarterly Distribution
Amount.  Distribution to Unitholders of a Special Distribution Amount will be
made no later than 15 days after the Special Distribution Amount record date.

Item 2.       Trustee's Discussion and Analysis of Financial Condition and
              Results of Operations

       The Trust makes quarterly cash distributions to holders of units of
beneficial interest ("Units") in the Trust ("Unitholders").  The only assets of
the Trust, other than cash and cash equivalents being held for the payment of
expenses and liabilities and for distribution to Unitholders, are certain net
profits interests (the "Royalty Interests") in certain proved coal seam gas
properties located in the Fruitland coal formation underlying the Northeast
Blanco Unit in the San Juan Basin of New Mexico (the "Underlying Properties").
The Royalty Interests owned by the Trust burden the net revenue interest in the
Underlying Properties that is owned by Burlington Resources Oil & Gas Company
("BROG") and not the Trust.  Effective January 1, 1996, Meridian Oil Production
Inc. ("MOPI") was merged with and into Meridian Oil Inc. ("MOI"), a wholly-
owned subsidiary of Burlington Resources.  Effective July 11, 1996, MOI changed
its name to Burlington Resources Oil & Gas Company and Meridian Oil Trading
Inc. ("MOTI") and Meridian Oil Gathering Inc. ("MOGI"), both affiliates of MOI,
changed their names to Burlington Resources Trading Inc. ("BRTI") and
Burlington Resources Gathering Inc. ("BRGI"), respectively.  Accordingly, with
regard to the period prior to the date of such name changes, references in this
Form 10-Q to BROG refer to MOPI, references to BRTI refer to MOTI and
references to BRGI refer to MOGI.

       Distributable income of the Trust consists of the excess of royalty
income plus interest income over the general and administrative expenses of the
Trust.  Upon receipt by the Trust, royalty income is invested in short-term
investments in accordance with the Trust Agreement (as defined in Note 1 to the
condensed financial statements of the Trust appearing elsewhere in this Form
10-Q ("Note 1")) until its subsequent distribution to Unitholders.





                                       12
<PAGE>   13
       The amount of distributable income of the Trust for any quarter may
differ from the amount of cash available for distribution to Unitholders in
such quarter due to differences in the treatment of the expenses of the Trust
in the determination of those amounts.  The condensed financial statements of
the Trust are prepared on a modified cash basis pursuant to which the expenses
of the Trust are recognized when paid or reserves are established for them.
Consequently, the reported distributable income of the Trust for any quarter is
determined by deducting from the income received by the Trust the amount of
expenses paid by the Trust during such quarter.  The amount of cash available
for distribution to Unitholders, however, is determined in accordance with the
provisions of the Trust Agreement and reflects the income actually received by
the Trust less the amount of expenses actually paid by the Trust and adjustment
for changes in reserves for unpaid liabilities.  See Note 4 to the condensed
financial statements of the Trust appearing elsewhere in this Form 10-Q for
additional information regarding the determination of the amount of cash
available for distribution to Unitholders.

Three Months and Six Months Ended June 30, 1998 Compared to Three Months and
Six Months Ended June 30, 1997

       Royalty income received by the Trust in a given calendar quarter
generally consists of 95% of the NPI Net Proceeds (as defined in Note 1) during
the preceding calendar quarter.  Royalty income for the second quarter of 1998
amounted to $2,028,502 as compared to $1,618,799 for the same quarter in 1997.
Gas production related to the royalty income received by the Trust in the
second quarter of 1998 was 3.0 Bcf compared to 2.6 Bcf for the same quarter in
1997.  The higher gas production in the second quarter of 1998 as compared to
the same period in 1997 is the result of improved looped gas gathering systems
installed in 1997.  The increase in royalty income to the Trust in the second
quarter of 1998 as compared to the same period in 1997 reflects the
aforementioned increase in production, although such increase was partially
offset by deductions for the recoupment of price credits as described herein
below.  The average net price for gas after consideration of costs in the
second quarter of 1998 was $.96/Mcf compared to $1.00/Mcf in the second quarter
of 1997.  Royalty Income for the six month period ended June 30, 1998 was
$3,452,856 compared to $3,090,719 for the same period in 1997. The higher
revenue for the six month period ended June 30, 1998 compared to the same
period June 30, 1997 is the result of the increased production as discussed
above.
        
       Costs deducted to determine the net proceeds received include production
taxes, which are calculated on gross proceeds, operating costs and certain
capital costs.  Production tax fluctuations are in correlation to changes in
royalty proceeds and operating costs and have remained relatively stable.
During the quarter ended June 30, 1998, approximately $436,000 was charged
against the gross proceeds of the Trust relating to the installation of low
pressure separators and the expansion of the current gas gathering system.
Capital costs of this nature are allowed under the Conveyance Agreement.

       Interest income for the quarter ended June 30, 1998 was $5,750 as
compared to $3,912 for the same quarter in 1997 due to more funds available for
investment.  Interest income for the six month period ended June 30, 1998 was
$10,199 compared to $7,589 for the same period for 1997.  General and
administrative expenses are primarily related to administrative services
provided to the Trust.  The general and administrative expenses during the
quarter ended June 30, 1998 amounted to $222,080 compared to $222,234 for the
same quarter in 1997.  The general and





                                       13
<PAGE>   14
administrative expenses for the six month period ended June 30, 1998 were
$408,273 compared to $418,520 for the same period for 1997.

       Distributable income for the quarter ended June 30, 1998 was $1,812,173
or $.21 per Unit compared to $1,400,477 or $.16 per Unit for the second quarter
of 1997.  The Trust made a distribution on June 12, 1998 of $.210321 per Unit
to Unitholders of record on June 2, 1998.  Distributable Income for the six
month period ended June 30, 1998 was $3,054,782 compared to $2,679,788 for the
same period in 1997.

       Royalty income to the Trust is attributable to the sale of depleting
assets.  All of the Underlying Properties burdened by the NPI consist of
producing properties.  Accordingly, the proved reserves attributable to BROG's
interest in the Underlying Properties are expected to decline substantially
during the term of the Trust and a portion of each cash distribution made by
the Trust will, therefore, be analogous to a return of capital.  Accordingly,
cash yields attributable to the Units are expected to decline over the term of
the Trust.

       Royalty income received by the Trust in a given calendar quarter will
generally reflect the proceeds from the sale of gas produced from the
Underlying Properties during the preceding quarter.  Accordingly, the royalty
income included in distributable income for the quarter ended June 30, 1998 was
based on production volumes and natural gas prices for the period January 1,
1998 through March 31, 1998 in accordance with the terms of the conveyance of
the Royalty Interests to the Trust, as shown in the table below.  The
production volumes included in the table below are actual net production
volumes from BROG's interest in the Underlying Properties, and not for
production attributable to the Trust's Royalty Interests.

<TABLE>
<CAPTION>
                                                    For the            For the
                                                  Three Months      Three Months
                                                      Ended             Ended
                                                  March 31, 1998   March 31, 1997
                                                  --------------   --------------
<S>                                                <C>              <C>         
Production (Bcf)(1).............................        3.197            2.749
Production (Trillion Btu)(2)....................        2.836            2.477
Average Inside FERC Price
       ($/MMBtu)(3).............................   $     1.95        $    2.71
BROG Average Entitled Price
       Received ($/MMBtu)(4)....................   $     1.08        $    1.09
</TABLE>

(1)    Billion Cubic Feet of natural gas.
(2)    Trillion British Thermal Units.
(3)    The posted index price (Inside FERC) of spot gas delivered to pipelines.
(4)    Average Inside FERC Price less allowable deductions.

       Production attributable to BROG's interest in the Underlying Properties
is generally sold pursuant to a gas purchase contract between BROG and BRTI.
The gas purchase contract





                                       14
<PAGE>   15
provides certain protections for BRTI in the form of price credits and for
Unitholders when the applicable Blanco Hub Spot Price falls below $1.65 per
MMBtu and provides certain benefits for BRTI when the Blanco Hub Spot Price
exceeds $2.10 per MMBtu.  The gas purchase contract also provides that the
price paid for gas by BRTI is reduced by the amount of gathering and/or
transportation charges, taxes, treating and processing costs and all other
costs payable in connection with such services from the central gathering point
to main line delivery paid by BRTI.

       The Blanco Hub Spot Price was above $1.65 per MMBtu during 1997 except
for March and April of 1997, and was also above such price throughout the first
and second quarters of 1998.  The price remained above $1.65 per MMBtu for July
1998. Historically, the Blanco Hub Spot Price has periodically been below $1.65
per MMBtu, however, pursuant to the terms of the gas purchase contract, in
those months in which such price was below $1.65 per MMBtu, BRTI continued to
purchase gas attributable to BROG's interest in the Underlying Properties at
the $1.60 per MMBtu minimum purchase price, less deductible costs paid by BRTI,
established by the gas purchase contract; and BRTI received a price credit from
BROG for each MMBtu of natural gas so purchased by BRTI equal to the difference
between the $1.60 per MMBtu purchase price and the applicable index price
(which price is equal to 97 percent of the applicable Blanco Hub Spot Price).
With the Blanco Hub Spot Price above $1.65 per MMBtu during January, February
and March 1998, royalty income otherwise receivable by the Trust during second
quarter of 1998 was reduced due to partial recoupment of the aggregate price
credits.  Approximately $611,000 of price credits were recouped by BRTI due to
the higher Blanco Hub Spot Prices in January, February and March of 1998.  BRTI
estimates that as of June 30, 1998, BRTI had aggregate price credits of
approximately $4.3 million of which the Trust's 95 percent interest was
approximately $4.1 million.

       The entitlement of BRTI to recoup the price credits means that if and
when the applicable Blanco Hub Spot Price is above $1.65 per MMBtu, future
royalty income paid to the Trust will be reduced until such time as all accrued
and unrecouped price credits have been recovered by BRTI.  As was the case in
the second quarter of 1998, reduced royalty income correspondingly reduces cash
distributions to Unitholders.

       The information in this Form 10-Q concerning production and prices
relating to BROG's interest in the Underlying Properties is based on
information prepared and furnished by BROG to the Trustee.  The Trustee has no
control over and no responsibility relating to the operation of the Underlying
Properties.

Year 2000

       Many existing computer programs use only two digits to identify a year
in the date field.  These programs were designed and developed without
considering the impact of the upcoming change in the century.  If not
corrected, many computer applications could fail or create erroneous results by
or at the Year 2000.  The Year 2000 issue affects virtually all companies and
organizations.  If a company or organization does not successfully address its
Year 2000 issues, it may face material adverse consequences.





                                       15
<PAGE>   16
       The Trust is reliant on the performance of BROG and third party vendors
for the receipt of Royalty income, payment of expenses and disbursement of
distributable income.  The Trustee can provide no assurance as to whether BROG
and third party vendors will successfully address the Year 2000 issue.  Any
failure by BROG or any of the third party vendors to successfully address the
Year 2000 issue could have a material adverse impact on the Trust and its
Unitholders.

Forward-Looking Statements

       This report on Form 10-Q includes "forward-looking statements" within
the meaning of Section 27A of the Securities Act of 1933, as amended, and
Section 21E of the Securities Exchange Act of 1934, as amended.  All statements
other than statements of historical fact included in this Form 10-Q, including,
without limitation, statements contained in this "Trustee's Discussion and
Analysis of Financial Condition and Results of Operations" regarding the
Trust's financial position and industry conditions, are forward-looking
statements.  Although the Trustee believes that the expectations reflected in
such forward-looking statements are reasonable, it can give no assurance that
such expectations will prove to have been correct.

Item 3.       Quantitative and Qualitative Disclosures About Market Risk.

       Not applicable.



                                       16
<PAGE>   17
                          PART II - OTHER INFORMATION


Item 5.       Other Information.

Unit Acquisitions by Certain Beneficial Owners

       On January 20, 1998, those certain parties listed in the Security
Ownership of Certain Beneficial Owners table (the "Ownership Table") set forth
in Item 12 of the Trust's Annual Report on Form 10-K for the year ended
December 31, 1997 (the "1997 Form 10-K") (such parties, collectively, "San
Juan") filed a Schedule 14D-1 with the Securities and Exchange Commission (the
"Commission") in connection with a tender offer to purchase exactly (and not
less than) 5,446,860 Units. The tender offer expired pursuant to its terms on
February 17, 1998.

       According to Schedule 13D filings made by San Juan with the Commission,
following the expiration of its tender offer, San Juan has acquired through a
series of open market transactions additional Units resulting in total holdings
as of July 10, 1998, of 5,331,468 Units, representing approximately 60.58% of
all outstanding Units. According to the Offer to Purchase filed by San Juan as
an exhibit to its Schedule 14D-1, the purpose of San Juan's tender offer was to
acquire 67% of the outstanding Units so that it could affect a vote to
terminate the Trust, causing a liquidation of the Trust's assets and a
resulting distribution to Unitholders.  Unitholders are advised to consult the
1997 Form 10-K, particularly "Item 1--Description of the Trust--Termination and
Liquidation" for additional information in this regard.

Merger of Trustee

       Effective May 6, 1998, NationsBank of Texas, N.A., of Dallas, Texas,
merged with NationsBank, N.A., of Charlotte, North Carolina. NationsBank, N.A.,
as survivor of the merger, has assumed by operation of law all rights,
properties and interests of NationsBank of Texas, N.A., including all rights and
interests as trustee, executor or administrator or in other fiduciary
capacities, in the same manner and to the same extent as such rights, properties
and interests were held by NationsBank of Texas, N.A. at the time of the merger.
In accordance with applicable law and the provisions of the Trust Agreement,
NationsBank, N.A., as successor to NationsBank of Texas, N.A., now serves as
Trustee of the Trust. NationsBank, N.A. is performing such functions through its
Dallas, Texas, branch office, which is the former principal office of
NationsBank of Texas, N.A.

Item 6.       Exhibits and Reports on Form 8-K.

       (a)    Exhibit No.                         Description
              -----------                         -----------

                   27                        Financial Data Schedule

       (b)    Reports on Form 8-K.

              No reports on Form 8-K were filed during the quarter for which
              this report is filed.





                                       17
<PAGE>   18
                                   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.


                                   BURLINGTON RESOURCES COAL SEAM GAS
                                   ROYALTY TRUST

                                   By: NationsBank, N.A., Trustee



                                   By: /s/ Ron Hooper                           
                                       -----------------------------------------
                                       Ron Hooper
                                       Vice President

Date:  August 12, 1998


               (The Trust has no directors or executive officers.)





                                       18

<PAGE>   19


                               INDEX TO EXHIBITS

<TABLE>
<CAPTION>
EXHIBIT
NUMBER              DESCRIPTION
-------             -----------
<S>                 <C>
 27                 Financial Data Schedule
</TABLE>