<PAGE>   1
                                                                   EXHIBIT 10.34


                   [DONALDSON, LUFKIN & JENRETTE LETTERHEAD]



                                                                January 16, 1998



PRIVATE AND CONFIDENTIAL

DLJ Merchant Banking II, Inc.
277 Park Avenue
New York, NY 10172

Attention:     Mr. Peter T. Grauer
               Managing Director

Ladies and Gentlemen:

     This letter agreement (the "Agreement") confirms our understanding that
DLJ Merchant Banking II, Inc. (the "Company") has engaged Donaldson, Lufkin &
Jenrette Securities Corporation ("DLJ") to (i) act as its exclusive financial
advisor for a period of 12 months commencing upon your acceptance of this
Agreement, with respect to the possible acquisition (and any related matters
such as financings) of Thermadyne Holdings Corporation ("Target"), in one or a
series of transactions, by merger, consolidation or any other business
combination, by purchase involving all or a substantial amount of the business,
securities or assets of the Target, or otherwise (each a "Acquisition"), and
(ii) following the closing of the Acquisition, and continuing for a period of
five years thereafter (the "Engagement Period"), with respect to the review and
analysis of financial and structural alternatives available to the Company with
a view to meeting its long term strategic objectives. It is understood and
agreed that the obligations of the Company under this letter agreement
(including without limitation, the indemnification and other obligations of the
Company set forth in Schedule I hereto) shall terminate upon the assumption of
such obligations by the Target upon the consummation of the Acquisition.
Accordingly, following consummation of the Acquisition, all references to the
Company shall be deemed a reference to the Target.

     As discussed, we propose to undertake certain services on your behalf, to
the extent requested by you, which shall consist of the following: (i)
assisting you in evaluating the Target, its operations, its historical
performance and its future prospects; (ii) advising on a proposed purchase
price and form of consideration; (iii) assisting you in structuring the
Acquisition; (iv) negotiating the financial aspects of any Acquisition under
your guidance and (v) assisting you, from time to time, in analyzing the
Company's operations, its historical performance and future prospects.

     As compensation for the services to be provided by DLJ hereunder, the
Company agrees (i) to pay to DLJ (a) a fee of $4,000,000 upon consummation of an
Acquisition (the "Acquisition Fee") and (b) an annual advisory fee of $300,000,
payable quarterly in equal installments of $75,000 on the first business day of
each three month period commencing on the closing of the Acquisition, and (ii)
upon request by DLJ from time to time, to reimburse DLJ promptly for all
out-of-pocket expenses (including the reasonable fees and expenses of counsel)
incurred by DLJ in connection with its engagement hereunder, whether or not an
Acquisition or a Transaction (as defined below) is consummated. As DLJ will be
acting on your behalf, the Company agrees to the
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Mr. Peter T. Grauer
DLJ Merchant Banking II, Inc.
Page 2                                                          January 16, 1998



indemnification and other obligations set forth in Schedule I attached hereto,
which Schedule is an integral part hereof.

     The Acquisition Fee shall be in an amount equal to $4,000,000 and shall be
payable in cash promptly upon consummation of the Acquisition. For purposes
of this Agreement, an Acquisition shall be deemed to have been consummated
upon the earliest of any of the following events to occur: (a) the acquisition
by the Company or any of its affiliates of at least fifty one percent (51.0%)
of the outstanding equity securities of the Target, calculated on a
fully-diluted basis; (b) a merger or consolidation of the Target with the
Company or an affiliate of the Company; (c) the acquisition by the Company or
any of its affiliates of assets of the Target representing at least fifty one
percent (51.0%) of the Target's book value or (d) in the case of any other
Acquisition, the consummation thereof.

     In the event an Acquisition is not consummated and the Company receives
cash or assets from the Target (such cash or assets hereinafter referred to as
the "Break-up Fee"), then the Company shall pay to DLJ in cash, promptly upon
the Company's receipt of such Break-up Fee, an amount equal to the lesser of
(x) $2,000,000 or (y) fifteen percent (15%) of such Break-up Fee received (the
"DLJ Participation").

     In connection with any debt or equity financing (other than bank
financing) by the Company (i) the proceeds of which are used to finance all or
a portion of the Acquisition or (ii) which is effected at any time within 12
months following consummation of the Acquisition, DLJ shall have the right but
not the obligation to act as exclusive private placement agent, lead initial
purchaser or sole managing underwriter to the Company, provided that the fees
of DLJ for such services shall be competitive with those customarily charged by
DLJ and other investment banks in similar transactions. In connection with such
financings, the Company shall take such action as is necessary or desirable to
facilitate the public or private sale of such securities, including the
preparation of any documents required to be filed with the Securities and
Exchange Commission and shall enter into a private placement agreement,
purchase agreement or underwriting agreement with DLJ which shall contain
normal and customary provisions for such agreements in which DLJ acts as
placement agent, initial purchaser or managing underwriter, as the case may be.

     As further compensation, the Company agrees that in the event the Company
determines to pursue any Transaction (as hereinafter defined) during the
Engagement Period, DLJ shall have the right to act as the Company's exclusive
financial advisor, sole placement agent, sole initial purchaser, sole managing
underwriter or sole dealer-manager, as the case may be, with respect to each
such Transaction.

     For purposes of this letter, the term "Transaction" shall specifically
exclude the Acquisition and shall include each of the following: (i) the sale,
merger, consolidation or any other business combination, in one or a series of
transactions, involving any portion of the business, securities or assets of
the Company; (ii) the acquisition (and any related matters such as financings,
divestitures, etc.), in one or a series of transactions, of all or a portion of
the business, securities or assets of another entity or person; (iii) any
recapitalization, refinancing, repurchase or restructuring of the Company's
equity or debt securities or indebtedness or any amendments or modifications to
the Company's debt securities or indentures whether or not in connection
therewith, involving, by or on behalf of the Company, an offer to purchase or
exchange for cash, property, securities, indebtedness or other consideration,
or a solicitation of consents, waivers of authorizations with respect thereto;
(iv) any spin-off, split-off or other extraordinary dividend of cash,
securities or other assets to
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Mr. Peter T. Grauer
DLJ Merchant Banking II, Inc.
Page 3                                                          January 16, 1998


stockholders of the Company: or (v) any sale of securities of the Company
effected pursuant to a private sale or an underwritten public offering.

     If the Company determines to pursue any such Transaction, DLJ and the
Company will enter into an agreement appropriate to the circumstances,
containing provisions for, among other things, compensation, indemnification,
contribution and representations and warranties, which are usual and customary
for similar agreements entered into by DLJ or other investment bankers of
international standing acting in similar transactions. DLJ shall have no
obligation to act as placement agent, initial purchaser, underwriter or dealer
manager to the Company or to place or purchase any securities of the Company,
except to the extent that such obligations arise out of a placement agent
agreement, purchase agreement, underwriting agreement or dealer-manager
agreement, as the case may be, with respect to a particular Transaction executed
and delivered by both DLJ and the Company.

     The Company shall make available to DLJ all financial and other information
concerning its business and operations that DLJ reasonably requests as well as
any other information relating to the Acquisition and any Transaction prepared
by the Company or any of its other advisors. In performing its services
hereunder, DLJ shall be entitled to rely without investigation upon all
information that is available from public sources as well as all other
information supplied to it by or on behalf of the Company or its advisors or the
Target or its advisors and shall not in any respect be responsible for the
accuracy or completeness of, or have any obligation to verify, the same or to
conduct any appraisal of any of the Company's or the Target's assets or
liabilities. To the extent consistent with legal requirements, all information
given to DLJ by the Company, unless publicly available or otherwise available to
DLJ without restriction or breach of any confidentiality agreement, will be held
by DLJ in confidence and will not be disclosed to anyone other than DLJ's agents
and advisors without the Company's prior approval or used for any purpose other
than those referred to in this Agreement.

     Any advice, written or oral, provided by DLJ pursuant to this Agreement
will be treated by the Company as confidential, will be solely for the
information and assistance of the Company in connection with its consideration
of the Acquisition or a Transaction and will not be reproduced, summarized,
described or referred to, or furnished to any other party or used for any other
purpose, except in each case with our prior written consent.

     The Company acknowledges and agrees that DLJ has been retained solely to
provide the advice or services set forth in this Agreement. DLJ shall act as an
independent contractor, and any duties of DLJ arising out of its engagement
hereunder shall be owed solely to the Company.

     This Agreement may be terminated by DLJ at any time or by the Company upon
expiration of the Engagement Period upon receipt of written notice to that
effect by the other party. Upon any termination or expiration of this Agreement,
DLJ will be entitled to prompt payment of all fees accrued prior to such
termination or expiration and reimbursement of all out-of-pocket expenses as
described above. The indemnity and other provisions contained in Schedule I will
also remain operative and in full force and effect regardless of any termination
or expiration of this Agreement.

     In addition, if at any time prior to 12 months after termination or
expiration of this Agreement an Acquisition is consummated or the Company is
entitled to receive a Break-up Fee, DLJ will be entitled to payment in full of
the Acquisition Fee or the DLJ Participation, as the case may be. Further, if at
any time prior to 12 months after the termination or expiration of this
                                 
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Mr. Peter T. Grauer
DLJ Merchant Banking II, Inc.
Page 4                                                         January 16, 1998



Agreement, a Transaction is consummated with respect to which DLJ was not
invited by the Company to act as financial advisor, placement agent, initial
purchaser, underwriter or dealer-manager, as the case may be, the Company
shall, promptly upon consummation of each such Transaction pay DLJ a cash fee
in an amount equal to the usual and customary fees which would have been to DLJ
by the Company if DLJ had acted in such capacity, with regard to the
consummated Transaction(s).

     It is understood that if the Company completes a transaction in lieu of
the Acquisition or any Transaction, either during the Engagement Period or at
any time prior to 12 months after termination by the Company or expiration of
this Agreement, for which DLJ is entitled to compensation pursuant to this
Agreement, DLJ and the Company will in good faith mutually agree upon
acceptable compensation for DLJ taking into account, among other things, the
results obtained and the custom and practice of investment bankers of
international standing acting in similar transactions.

     The Company further agrees that it will not enter into any transaction
referred to in either of the two preceding paragraphs unless, prior to or
simultaneously with such transaction, adequate provision is made with respect
to the payment of compensation to DLJ as contemplated by such paragraphs.

     This Agreement shall be binding upon and inure to the benefit of the
Company, DLJ, each Indemnified Person (as defined in Schedule I) and their
respective successors and assigns.

     This Agreement shall be governed by, and construed and enforced in
accordance with, the laws of the State of New York.

     The Company irrevocably and unconditionally submits to the exclusive
jurisdiction of any State or Federal court sitting in New York city over any
suit, action or proceeding arising out of or relating to this Agreement
(including Schedule I). The Company hereby agrees that service of any process,
summons, notice or document by U.S. registered mail addressed to the Company
shall be effective service of process for any action, suit or proceeding brought
in any such court. The Company irrevocably and unconditionally waives any
objection to the laying of venue of any such suit, action or proceeding
brought in any such court and any claim that any such suit, action or
proceeding brought in such a court has been brought in an inconvenient forum.
The Company agrees that a final judgment in any such suit, action or proceeding
brought in any such court shall be conclusive and binding upon the Company and
may be enforced in any other courts to whose jurisdiction the Company is or may
be subject, by suit upon such judgment.


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Mr. Peter T. Grauer
DLJ Merchant Banking II, Inc.
Page 5
                                                                January 16, 1998


     If any term, provision, covenant or restriction contained in this
Agreement, including Schedule I, is held by a court of competent jurisdiction
or other authority to be invalid, void, unenforceable or against its regulatory
policy, the remainder of the terms, provisions, covenants and restrictions
contained in this Agreement shall remain in full force and effect and shall in
no way be affected, impaired or invalidated.

     After reviewing this Agreement, please confirm that the foregoing is in
accordance with your understanding by signing and returning to me the duplicate
of this Agreement attached hereto, whereupon it shall be our binding Agreement.

                                             Very truly yours,


                                             DONALDSON, LUFKIN & JENRETTE
                                              SECURITIES CORPORATION

                                             By: /s/ JOSEPH H. MARREN
                                                --------------------------------
                                                Joseph H. Marren
                                                Senior Vice President



Accepted and Agreed
this       day of           1998
     -----       ----------

DLJ MERCHANT BANKING II, INC.

By: /s/ PETER T. GRAUER
   -----------------------
   Peter T. Grauer
   Managing Director
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                                   SCHEDULE I

     This Schedule I is a part of and is incorporated into that certain letter
agreement (together, the "Agreement") dated January 16, 1998 by and between DLJ
Merchant Banking II, Inc. (the "Company") and Donaldson, Lufkin & Jenrette
Securities Corporation ("DLJ").

     The Company will indemnify and hold harmless DLJ and its affiliates, and
the respective directors, officers, agents and employees of DLJ and its
affiliates (DLJ and each such entity or person, an "Indemnified Person") from
and against any losses, claims, damages, judgments, assessments, costs and
other liabilities (collectively "Liabilities"), and will reimburse each
Indemnified Person for all fees and expenses (including the reasonable fees and
expenses of counsel) (collectively "Expenses") as they are incurred in
investigating, preparing, pursuing, or defending any claim, action, proceeding
or investigation, whether or not in connection with pending or threatened
litigation or arbitration and whether or not any Indemnified Person is a party
(collectively, "Actions"), arising out of or in connection with advice or
services rendered or to be rendered by any Indemnified Person pursuant to this
Agreement, the transactions contemplated hereby or any Indemnified Person's
actions or inactions in connection with any such advice, services or
transactions; provided that the Company will not be responsible for any
Liabilities or Expenses of any Indemnified Person that are determined by a
judgment of a court of competent jurisdiction which is no longer subject to
appeal or further review to have resulted solely from such Indemnified Person's
gross negligence or willful misconduct in connection with any of the advice,
actions, inactions or services referred to above; it being understood and
agreed that for purposes of this Schedule I, the term "Indemnified Person"
shall specifically exclude the Company.  The Company also agrees to reimburse
each Indemnified Person for all Expenses as they are incurred in connection
with enforcing such Indemnified Person's rights under this Agreement
(including, without limitation, its rights under this Schedule I).

     Upon receipt by an Indemnified Person of actual notice of an Action
against such Indemnified Person with respect to which indemnity may be sought
under this Agreement, such Indemnified Person shall promptly notify the Company
in writing; provided that failure so to notify the Company shall not relieve
the Company from any liability which the Company may have on account of this
indemnity or otherwise, except to the extent the Company shall have been
materially prejudiced by such failure.  The Company shall, if requested by DLJ,
assume the defense of any such Action including the employment of counsel
reasonably satisfactory to DLJ.  Any Indemnified Person shall have the right to
employ separate counsel in any such action and participate in the defense
thereof but the fees and expenses of such counsel shall be at the expense of
such Indemnified Person, unless: (i) the Company has failed promptly to assume
the defense and employ counsel or (ii) the named parties to any such Action
(including any impleaded parties) include such Indemnified Person and the
Company, and such Indemnified Person shall have been advised by counsel that
there may be one or more legal defenses available to it which are different
from or in addition to those available to the Company; provided that the
Company shall not in such event be responsible hereunder for the fees and
expenses of more than one firm of separate counsel in connection with any
Action in the same jurisdiction, in addition to any local counsel.  The Company
shall not be liable for any settlement of any Action effected without its
written consent.  In addition, the Company will not, without prior written
consent of DLJ, settle, compromise or consent to the entry of any judgment in
or otherwise seek to terminate any pending or threatened Action in respect
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of which indemnification or contribution may be sought hereunder (whether or
not any Indemnified Person is a party thereto) unless such settlement,
compromise, consent or termination includes an unconditional release of each
Indemnified Person from all Liabilities arising out of such Action.

     In the event the foregoing indemnity is unavailable to an Indemnified
Person other than in accordance with this Agreement, the Company shall
contribute to the Liabilities and Expenses paid or payable by such Indemnified
Person in such proportion as is appropriate to reflect (i) the relative
benefits to the Company and its shareholders, on the one hand, and to DLJ, on
the other hand, of the matters contemplated by this Agreement or (ii) if the
allocation provided by the immediately preceding clause is not permitted by the
applicable law, not only such relative benefits but also the relative fault of
the Company, on the one hand, and DLJ, on the other hand, in connection with
the matters as to which such Liabilities or Expenses relate, as well as any
other relevant equitable considerations; provided that in no event shall the
Company contribute less than the amount necessary to ensure that all
Indemnified Persons, in the aggregate, are not liable for any Liabilities and
Expenses in excess of the amount of fees actually received by DLJ pursuant to
this Agreement. For purposes of this paragraph, the relative benefits to the
Company and its shareholders, on the one hand, and to DLJ, on the other hand,
of the matters contemplated by this Agreement shall be deemed to be in the same
proportion as (a) the total value paid or contemplated to be paid or received
or contemplated to be received by the Company or the Company's shareholders, as
the case may be, in the transaction or transactions that are within the scope
of this Agreement, whether or not any such transaction is consummated, bears to
(b) the fees paid or to be paid to DLJ under this Agreement.

     The Company also agrees that no Indemnified Person shall have any
liability (whether direct or indirect, in contract or tort or otherwise) to the
Company for or in connection with advice or services rendered or to be rendered
by any Indemnified Person pursuant to this Agreement, the transactions
contemplated hereby or any Indemnified Person's actions or inactions in
connection with any such advice, services or transactions except for Liabilities
(and related Expenses) of the Company that are determined by a judgment of a
court of competent jurisdiction which is no longer subject to appeal or further
review to have resulted solely from such Indemnified Person's gross negligence
or willful misconduct in connection with any such advice, actions, inactions or
services.

     The reimbursement, indemnity and contribution obligations of the Company
set forth herein shall apply to any modification of this Agreement and shall
remain in full force and effect regardless of any termination of, or the
completion of any Indemnified Person's services under or in connection with,
this Agreement.