Document:

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                                                                    EXHIBIT 10.4

                     FIRST AMENDMENT TO EMPLOYMENT AGREEMENT

         THIS FIRST AMENDMENT TO EMPLOYMENT AGREEMENT (this "First Amendment")
dated as of April 26, 2001 (the "Amendment Effective Date") by and between The
Houston Exploration Company, a Delaware corporation (the "Company"), and Thomas
E. Schwartz (the "Executive").

                                   WITNESSETH:

         WHEREAS, the Company and the Executive entered into an Employment
Agreement dated as of May 1, 1998 (the "Agreement"); and

         WHEREAS, the Company and the Executive hereby desire to amend the
Agreement as set forth herein;

         NOW, THEREFORE, in consideration of the premises and for other good and
valuable consideration, the receipt and sufficiency of which are hereby
acknowledged, the Company and the Executive hereby agree as follows:

1. The Agreement shall be amended to reflect that the Executive's position with
the Company shall be that of Vice President, Geophysics.

2. The Agreement shall be amended by adding the following provisions to the end
of Section 1:

                  The Executive agrees to continue his employment with the
                  Company. Consequently, the Executive's employment with the
                  Company shall continue, and this Agreement shall remain in
                  full force and effect according to its terms.

3. The Agreement shall be amended by changing the salary per year referenced in
Section 3 from "$160,000 per year" to "$190,000 per year."

4. The Agreement shall be amended by deleting the language in Section 7(e) after
paragraph (iii) thereof in the definition of the term "Good Reason" continued in
subparts (G)(1)-(4), which formerly required that certain events also occur
before a Change of Control would constitute Good Reason and trigger the lump sum
severance payments provided in Section 7(e). As amended, subpart (G) of the
definition of Good Reason now reads in its entirety as follows:

                  (G)      the occurrence of a Change of Control.

5. The Agreement shall be amended by modifying the second sentence of Section
6(a) to provide that the termination of the Agreement will also terminate the
non-compete provisions contained therein. As amended, Section 6(a) now reads in
its entirety as follows:

                  (a) Noncompetition Activities. The Executive acknowledges that
                  the nature of the employment under this Agreement is such as
                  will bring the Executive in personal contact with patrons or
                  customers of the Company and will enable him to acquire
                  valuable information as to the nature and character of the
                  business of the Company, thereby enabling him, by engaging in
                  the competing business on his own behalf, or for another, to
                  take advantage of such knowledge and thereby gain an unfair
                  advantage. Accordingly, the Executive covenants and agrees
                  that he will not, without the prior written consent of the
                  Company during the Term of Employment, engage directly or
                  indirectly for himself, or as an agent, representative,
                  officer, director or employee of others, in the exploration
                  for or production of hydrocarbons in waters offshore from the

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                  States of Texas and Louisiana, provided that the foregoing
                  restriction shall not apply at any time after the Executive's
                  Term of Employment and, provided further, that nothing in this
                  Agreement shall prohibit the Executive from acquiring or
                  holding any issue of stock or securities of any entity
                  registered under Section 12 of the Securities and Exchange Act
                  of 1934 (as amended), listed on a national securities exchange
                  or quoted on the automated quotation system of the National
                  Association of Securities Dealers, Inc., so long as the
                  Executive is not deemed to be an "affiliate" of such entity,
                  as such term is used in paragraphs (c) and (d) of Rule 145
                  under the Securities Act of 1933 (as amended).

6. The Agreement shall be amended to clarify the scope of the term "total
compensation" in Section 7(e). Section 7(e)(i) shall be amended to read in its
entirety as follows:

         (i) pay to the Executive, within thirty (30) days after the date of
such termination, a lump sum payment equal to 2.99 times the Executive's
then-current annual rate of Total Compensation;

The following provision shall be added to the end of Section 7(e):

            As used in this Agreement, the term "Total Compensation" shall mean
the sum of the following:

         (i) the current annual salary of the Executive referenced in Section 3;

         (ii) the current car allowance provided by the Company to the Executive
referenced in Section 4(c); and

         (iii) the Executive's annual bonus, calculated as though the Company's
financial targets had been met at one hundred percent (100%) referenced in
Section 3 and Exhibit A hereto.

7. Exhibit A to the Agreement shall be amended by changing the "Percentage of
Salary for Target Annual Bonus" for the Executive from 45% to 55%.

8. The Agreement shall be amended by modifying the last sentence of Section 14
to read in its entirety as follows:

                  This Agreement, as amended by the First Amendment, constitutes
                  the sole agreement between the parties with respect to the
                  employment of the Executive by the Company and supersedes any
                  and all other agreements, oral or written, between the
                  parties.

9. Except as expressly amended hereby, the Company and the Executive ratify and
confirm all terms and conditions of the Agreement as continuing in full force
and effect.

                                       -2-
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         IN WITNESS WHEREOF, the parties hereto have duly executed this First
Amendment as of the date first above written.

                                   THE HOUSTON EXPLORATION COMPANY

Addresses:

1100 Louisiana, Suite 2000
Houston, Texas 77002               By: /s/ William G. Hargett
                                      -----------------------
                                   Name: William G. Hargett
                                   Title: President and Chief Executive Officer

100 Louisiana, Suite 2000
Houston, Texas 77002               By: /s/ Thomas E. Schwartz
                                      -----------------------
                                      Thomas E. Schwartz

                                       -3-<PAGE>   1
                                                                    EXHIBIT 10.5

                     FIRST AMENDMENT TO EMPLOYMENT AGREEMENT

         THIS FIRST AMENDMENT TO EMPLOYMENT AGREEMENT (this "First Amendment")
dated as of April 26, 2001 (the "Amendment Effective Date") by and between The
Houston Exploration Company, a Delaware corporation (the "Company"), and James
F. Westmoreland (the "Executive").

                                   WITNESSETH:

         WHEREAS, the Company and the Executive entered into an Employment
Agreement dated as of July 2, 1996 (the "Agreement") with an Effective Date of
September 19, 1996; and

         WHEREAS, the Company and the Executive hereby desire to amend the
Agreement as set forth herein;

         NOW, THEREFORE, in consideration of the premises and for other good and
valuable consideration, the receipt and sufficiency of which are hereby
acknowledged, the Company and the Executive hereby agree as follows:

1. The Agreement shall be amended to reflect that the Executive's position with
the Company shall be that of Vice President, Chief Accounting Officer,
Comptroller and Secretary.

2. The Agreement shall be amended by adding the following provisions to the end
of Section 1:

                  The Executive agrees to continue his employment with the
                  Company. Consequently, the Executive's employment with the
                  Company shall continue, and this Agreement shall remain in
                  full force and effect according to its terms.

3. The Agreement shall be amended by changing the salary per year referenced in
Section 3 from "$125,000 per year" to "$190,000 per year."

4. The Agreement shall be amended by deleting the language in Section 7(e) after
paragraph (iii) thereof in the definition of the term "Good Reason" continued in
subparts (G)(1)-(4), which formerly required that certain events also occur
before a Change of Control would constitute Good Reason and trigger the lump sum
severance payments provided in Section 7(e). As amended, subpart (G) of the
definition of Good Reason now reads in its entirety as follows:

                  (G)      the occurrence of a Change of Control.

5. The Agreement shall be amended by modifying the second sentence of Section
6(a) to provide that the termination of the Agreement will also terminate the
non-compete provisions contained therein. As amended, Section 6(a) now reads in
its entirety as follows:

                  (a) Noncompetition Activities. The Executive acknowledges that
                  the nature of the employment under this Agreement is such as
                  will bring the Executive in personal contact with patrons or
                  customers of the Company and will enable him to acquire
                  valuable information as to the nature and character of the
                  business of the Company, thereby enabling him, by engaging in
                  the competing business on his own behalf, or for another, to
                  take advantage of such knowledge and thereby gain an unfair
                  advantage. Accordingly, the Executive covenants and agrees
                  that he will not, without the prior written consent of the
                  Company during the Term of Employment, engage directly or
                  indirectly for himself, or as an agent, representative,
                  officer, director or employee of others, in the exploration
                  for or production of hydrocarbons in waters offshore from the

<PAGE>   2

                  States of Texas and Louisiana, provided that the foregoing
                  restriction shall not apply at any time after the Executive's
                  Term of Employment and, provided further, that nothing in this
                  Agreement shall prohibit the Executive from acquiring or
                  holding any issue of stock or securities of any entity
                  registered under Section 12 of the Securities and Exchange Act
                  of 1934 (as amended), listed on a national securities exchange
                  or quoted on the automated quotation system of the National
                  Association of Securities Dealers, Inc., so long as the
                  Executive is not deemed to be an "affiliate" of such entity,
                  as such term is used in paragraphs (c) and (d) of Rule 145
                  under the Securities Act of 1933 (as amended).

6. The Agreement shall be amended to clarify the scope of the term "total
compensation" in Section 7(e). Section 7(e)(i) shall be amended to read in its
entirety as follows:

         (i) pay to the Executive, within thirty (30) days after the date of
such termination, a lump sum payment equal to 2.99 times the Executive's
then-current annual rate of Total Compensation;

The following provision shall be added to the end of Section 7(e):

         As used in this Agreement, the term "Total Compensation" shall mean the
sum of the following:

         (i) the current annual salary of the Executive referenced in Section 3;

         (ii) the current car allowance provided by the Company to the Executive
referenced in Section 4(c); and

         (iii) the Executive's annual bonus, calculated as though the Company's
financial targets had been met at one hundred percent (100%) referenced in
Section 3 and Exhibit A hereto.

7. Exhibit A to the Agreement shall be amended by changing the "Percentage of
Salary for Target Annual Bonus" for the Executive from 45% to 55%.

8. The Agreement shall be amended by modifying the last sentence of Section 14
to read in its entirety as follows:

                  This Agreement, as amended by the First Amendment, constitutes
                  the sole agreement between the parties with respect to the
                  employment of the Executive by the Company and supersedes any
                  and all other agreements, oral or written, between the
                  parties.

9. Except as expressly amended hereby, the Company and the Executive ratify and
confirm all terms and conditions of the Agreement as continuing in full force
and effect.

                                       -2-
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         IN WITNESS WHEREOF, the parties hereto have duly executed this First
Amendment as of the date first above written.

                                   THE HOUSTON EXPLORATION COMPANY

Addresses:

1100 Louisiana, Suite 2000
Houston, Texas 77002               By: /s/ William G. Hargett
                                      -----------------------
                                   Name: William G. Hargett
                                   Title: President and Chief Executive Officer

1100 Louisiana, Suite 2000
Houston, Texas 77002               By: /s/ James F. Westmoreland
                                      --------------------------
                                      James F. Westmoreland

                                       -3-

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