<PAGE>   1
                                                                  Exhibit 10.3.4

                                  BANK ONE, NA
                   BUSINESS PURPOSE REVOLVING PROMISSORY NOTE
                                  (SWING LINE)

$20,000,000.00                                                       Akron, Ohio

                                                                  August 4, 1998

         FOR VALUE RECEIVED, the undersigned, TELXON CORPORATION, a Delaware
corporation (the "Borrower"), hereby promises to pay to the order of BANK ONE,
NA FKA BANK ONE, AKRON, NA (hereinafter called the "Bank," which term shall
include any holder hereof), at such place as the Bank may designate or, in the
absence of such designation, at any of the Bank's offices, the sum of Twenty
Million and No/100ths dollars ($20,000,000.00) or so much thereof as shall have
been advanced by the Bank at any time and not hereafter repaid, together with
interest calculated daily on the current outstanding principal balance at a
variable rate at the time of each request for a money advance under this Note
equal to the "Money Market rate", as determined solely by Bank, plus the
"Applicable Margin" as defined with respect to Eurodollar Advances in Section 1
of the Agreement referred to below. Interest shall be due monthly upon this Note
and shall be payable for each month on the fifteenth day of the next subsequent
month commencing on the fifteenth day of the August, 1998. If not sooner paid,
this Note shall mature and all principal, interest and unpaid expenses shall be
due and payable in full on August 3, 1999. Bank shall have the right to assess a
late payment processing fee in the amount of Fifty and no/100ths Dollars
($50.00) or five percent (5%) of the scheduled payment in the event of default
in payment that remains uncured for a period of at least ten (10) days. The
proceeds of the loan evidenced hereby may be advanced, repaid and re-advanced,
in partial amounts, until maturity. Each money advance may be made to the
Borrower during the term hereof, in the Bank's sole discretion, upon receipt by
the Bank of the Borrower's request therefor, which request shall be made in
accordance with reasonable procedures which Bank shall from time to time
prescribe. Telephonic requests received by appropriate Bank representatives
prior to 3:00 p.m. local time (Akron, Ohio) shall be funded the same business
day. Telephonic requests received after 3:00 p.m. local time may not be funded
until the next subsequent business day. The Bank shall be entitled to rely on
any oral or telephonic communication requesting a money advance and/or providing
disbursement instructions hereunder, which shall be received by it in good faith
from anyone reasonably believed by the Bank to be the Borrower, or the
Borrower's authorized agent. The Borrower agrees that all money advances made by
the Bank, and interest thereon, will be evidenced by entries made by the Bank to
a loan account through its electronic data processing system and/or internal
memoranda maintained by the Bank. The Borrower further agrees that the sum or
sums shown on loan account from the Bank's electronic data processing system
and/or such memoranda shall be conclusively binding evidence of the amount of
the principal sum and of the amount of any accrued interest, except as to
manifest errors.

         There shall be no penalty for prepayment. All payments shall be applied
in the following order (i) fees and expenses (including reasonable attorney
fees) incurred by Bank in connection with the enforcement of this Note, (ii)
accrued but unpaid interest, and last (iii) principal.


<PAGE>   2



         Borrower is a party to a Credit Agreement, dated March 8, 1996, as
amended by Amendment No. 1 dated as of August 6, 1996 and Amendment No. 2 dated
as of December 16, 1996, with The Bank of New York ("BNY") acting as agent,
issuer and swing line lender. Such amended agreement, as the same may be further
amended, modified or replaced from time to time, is hereafter referred to as the
"Agreement". Borrower agrees that so long as this Note remains in effect and
until paid in full, Borrower shall neither create nor permit any lien, mortgage,
deed of trust, or security interest in its property or assets other than those
granted or permitted (including under Section 8.2 thereof) under the terms of
the Agreement. Borrower further agrees that a default under the Agreement,
including but not limited to a default under the affirmative or negative
covenants under the Agreement (whether the default is declared or undeclared,
waived or not waived, by the lender under the Agreement), shall, at Bank's
option, constitute a default under this Note. Borrower shall not amend or modify
the material terms of the Agreement, without the Bank's consent. Borrower shall
notify Bank in writing within twenty-four hours of its receipt of notice of a
claimed default under the Agreement. The terms and conditions of the Agreement
with respect to events of defaults, notices and cure rights and remedies are
incorporated herein by reference to the extent applicable.

         Upon the occurrence of any one or more defaults under this Note, the
holder hereof, at its option, may declare the entire unpaid balance of principal
and interest on this Note to be immediately due and payable, without notice or
demand, and may, at its option, cumulatively exercise any other right or remedy
provided at law or equity. Failure to exercise any such option shall not
constitute a waiver of the right to exercise the same in the event of any
subsequent default. Upon Bank's declaration that the entire unpaid balance is
immediately due, the unpaid balance of principal shall bear interest at 2% plus
the rate otherwise applicable to this Note.

         All of the parties hereto, including the Borrower, and any endorser,
surety, or guarantor, hereby severally waive presentment, notice of dishonor,
protest, notice of protest, and diligence in bringing suit against any party
hereto, and consent that, without discharging any of them, the time of payment
may be extended an unlimited number of times before or after maturity by the
Bank without notice to them.

         The obligations evidenced hereby may from time to time be evidenced by
amendments hereto or another note or note given in substitution, renewal or
extension hereof.

         The obligations of the Borrower under this Note shall constitute
"Designated Senior Indebtedness" under, and as such term is defined and used as
of the date hereof in, the Indenture, dated as of December 1, 1995, between the
Borrower and Bank One Trust Company, N.A., as Trustee, as the same may be
amended, supplemented or otherwise modified from time to time, with respect to
the $82,500,000 in issued and outstanding principal amount of the Borrower's
5-3/4% Convertible Subordinated Notes due 2003.



<PAGE>   3


         This Note evidences the Credit Line under, and as such term is defined
in, the Intercreditor Agreement by and between Bank and BNY, as acknowledged by
Borrower, and is subject to, and should be construed in conjunction with, the
provisions thereof.

         If any terms or provisions of this Note shall be deemed unenforceable,
the enforceability of the remaining terms and provisions shall not be affected.
This Note shall be governed by and construed in accordance with the laws of the
State of Ohio.

                                    BORROWER:

                                    TELXON CORPORATION


                                     By: /s/ Kenneth W. Haver
                                        ----------------------------------------
                                     Kenneth W. Haver, Senior Vice President and
                                             Chief Financial Officer