<PAGE>   1
                                                                  EXHIBIT 3.1(c)


                   INTERNATIONAL REMOTE IMAGING SYSTEMS, INC.
        CERTIFICATE OF DESIGNATIONS OF SERIES B CALLABLE PREFERRED STOCK

                   INTERNATIONAL REMOTE IMAGING SYSTEMS, INC.


                           CERTIFICATE OF DESIGNATIONS
                                       OF
                        SERIES B CALLABLE PREFERRED STOCK

             (Pursuant to Section 151 of the General Corporation Law
                            of the State of Delaware)

     International Remote Imaging Systems, Inc., a Delaware corporation (the
"Corporation"), in accordance with the provisions of Section 103 of the General
Corporation Law of the State of Delaware (the "DGCL"), DOES HEREBY CERTIFY:

     That pursuant to authority vested in the Board of Directors of the
Corporation by the Certificate of Incorporation of the Corporation, the Board of
Directors of the Corporation, at a meeting held on June 11, 1999, adopted a
resolution providing for the creation of a series of the Corporation's Preferred
Stock, $.01 par value, which series is designated "Series B Callable Preferred
Stock", which resolution is as follows:

     RESOLVED, that pursuant to authority vested in the Board of Directors of
the Corporation by the Certificate of Incorporation, the Board of Directors does
hereby provide for the creation of a series of the Preferred Stock, $.01 par
value (hereafter called the "Preferred Stock"), of the Corporation, and to the
extent that the voting powers and the designations, preferences and relative,
participating, optional or other special rights thereof and the qualifications,
limitations or restrictions of such rights have not been set forth in the
Certificate of Incorporation, as amended, of the Corporation, does hereby fix
the same as follows:


                        SERIES B CALLABLE PREFERRED STOCK

     SECTION 1. DESIGNATION AND AMOUNT. The shares of such series shall be
designated as "Series B Callable Preferred Stock" (the "Series B Preferred
Stock"), and the number of shares constituting the Series B Preferred Stock
shall be 256,000, and shall not be subject to increase.

     SECTION 2. STATED CAPITAL. The amount to be represented in stated capital
at all times for each share of Series B Preferred Stock shall be $3.00.

     SECTION 3. RANK. All Series B Preferred Stock shall rank (i) senior to the
Common Stock, $.01 par value (the "Common Stock"), of the Corporation, now or
hereafter issued, as to payment or distribution of assets upon liquidation,
dissolution or winding up of the Corporation, whether voluntary or involuntary,
(ii) junior to the outstanding shares of Series A Convertible Preferred Stock,
$.01 par value (the "Series A Preferred Stock"), and (iii) on a parity with or
senior to any additional series of Preferred Stock which the Board of Directors
or the stockholders may from time to time authorize, as to distributions of
assets upon liquidation, dissolution, or winding up of the Corporation, whether
voluntary or involuntary.

     SECTION 4. DIVIDENDS AND DISTRIBUTIONS.

     (a)  The holders of shares of Series B Preferred Stock shall not be
entitled to receive any dividends, except as otherwise provided herein.

     (b)  Unless the Corporation shall have first obtained the written consent
of the holders of a majority of the Series B Preferred Stock at the time
outstanding, so long as any shares of Series B Preferred Stock are outstanding,
the Corporation shall not pay or declare and set apart for such payment any
dividend or distribution on shares of Common Stock or Junior Stock (as defined
herein) (other than (1) dividends on shares of Common Stock solely in the form
of additional shares of Common Stock or (2) dividends on Junior Stock solely in
the form of shares of Common Stock or additional shares of Junior Stock) or
recapitalize the shares of Common Stock into other securities or property or
change the par value thereof.

     (c)  Unless the Corporation shall have first obtained the written consent
of the holders of a majority of the Series B Preferred Stock at the time
outstanding, so long as any shares of Series B Preferred Stock are outstanding,

<PAGE>   2

neither the Corporation nor any subsidiary of the Corporation shall redeem,
repurchase or otherwise acquire in any one transaction or series of related
transactions any shares of Common Stock or Junior Stock if the aggregate
purchase price or cost of the shares so repurchased, redeemed or otherwise
acquired since the initial issuance of shares of Series B Preferred Stock (such
date of initial issuance being referred to herein as the "Issuance Date")
exceeds $100,000; provided, however, that this limitation shall not apply to the
exercise of contractual repurchase rights to repurchase (at prices below then
fair market value) shares of Common Stock previously sold to employees pursuant
to the Corporation's Key Employee Stock Purchase Plan or other employee stock
purchase or option plan adopted by the Board of Directors and approved by the
Corporation's stockholders.

     (d)  Unless the Corporation shall have first obtained the written consent
of the holders of a majority of the Series B Preferred Stock at the time
outstanding, so long as any shares of Series B Preferred Stock are outstanding,
neither the Corporation nor any subsidiary of the Corporation shall make any
tender offer or exchange offer (a "Tender Offer") for outstanding shares of
Common Stock.

     SECTION 5. LIQUIDATION PREFERENCE. In the event of a liquidation,
dissolution, or winding up of the Corporation, whether voluntary or involuntary,
the holders of Series B Preferred Stock shall be entitled to receive out of the
assets of the Corporation an amount per share of Series B Preferred Stock equal
to $3.00 (collectively, the "Liquidation Preference"), and no more, before any
payment shall be made or any assets distributed to the holders of Common Stock
or any other class or series of the Corporation's capital stock junior as to
liquidation or dividend rights to the Series B Preferred Stock (collectively,
the "Junior Stock"); provided, however, that such rights shall accrue to the
holders of Series B Preferred Stock only in the event that the Corporation's
payments with respect to the liquidation preference of the holders of capital
stock of the Corporation ranking senior as to liquidation rights to the Series B
Preferred Stock (the "Senior Liquidation Stock") are fully met. After the
liquidation preferences of the Senior Liquidation Stock are fully met, the
entire assets of the Corporation available for distribution shall be distributed
ratably among the holders of the Series B Preferred Stock and any other class or
series of the Corporation's capital stock having parity as to liquidation rights
with the Series B Preferred Stock (the "Parity Liquidation Stock") in proportion
to the respective preferential amounts to which each is entitled (but only to
the extent of such preferential amounts). After payment in full of the
liquidation price of the shares of the Series B Preferred Stock and the Parity
Liquidation Stock, the holders of such shares shall not be entitled to any
further participation in any distribution of assets by the Corporation.

     SECTION 6. NO MANDATORY REDEMPTION. The shares of Series B Preferred Stock
shall not be subject to mandatory redemption by the Corporation.

     SECTION 7. NO SINKING FUND. The shares of Series B Preferred Stock shall
not be subject to the operation of a purchase, retirement or sinking fund.

     SECTION 8. NO OPTIONAL CASH REDEMPTION. The shares of Series B Preferred
Stock shall not be subject to redemption for cash at the option of the
Corporation.

     SECTION 9. RIGHT OF CORPORATION TO CALL OUTSTANDING SHARES.

     (a)  CALL OPTION OF CORPORATION. The Corporation may at any time call for
conversion all (but not part) of the outstanding shares of Series B Preferred
Stock and convert them into fully paid and nonassessable shares of Common Stock
as hereinafter provided. The Company may convert each outstanding share of
Series B Preferred Stock into such number of fully paid and nonassessable shares
of Common Stock (calculated as to each conversion to the nearest 1/100th of a
share) determined by dividing (x) the Conversion Amount by (y) the arithmetic
average of the Closing Price of the Common Stock on the five consecutive trading
days ending three trading days before the Conversion Date (but in no event less
than $2.00 per share) (subject to equitable adjustments for stock splits, stock
dividends, combinations, recapitalizations, reclassifications and similar events
occurring on or after the date of filing of this Certificate of Designations
with the Secretary of State of the State of Delaware) (the "Conversion Price").
The "Conversion Rate" shall be equal to the Conversion Amount divided by the
Conversion Price, and the Conversion Price shall be equal to the Conversion
Amount divided by the Conversion Rate.

     (b)  MANNER OF EXERCISE. If the Corporation elects to exercise its call
option, it shall mail a written notice to each holder of record of the Series B
Preferred Stock by first class mail, postage prepaid, to the last address
indicated on the books of the Corporation. Any notice mailed in this manner
shall be conclusively presumed to have been duly given whether or not actually
received. The notice of call shall specify the Conversion Price, the date fixed
the exchange of certificates (such date to be not less than ten (10) nor more
than sixty (60) days after the date of such notice) and (iii) instructions for
the exchange of stock certificates. Promptly, but in no event later than ten
(10) business days after delivery of a call notice, each holder shall surrender
for such purpose to the Corporation or its agent certificates representing
shares which have been called, duly endorsed in blank or accompanied by proper
instruments of transfer if required. The Corporation shall pay any tax arising
in connection with any conversion of shares of Series B Preferred Stock except
that

<PAGE>   3

the Corporation shall not, however, be required to pay any tax which may be
payable in respect of any transfer involved in the issue and delivery upon
conversion of shares of Common Stock or other securities or property in a name
other than that of the holder of the shares of the Series B Preferred Stock
being converted, and the Corporation shall not be required to issue or deliver
any such shares or other securities or property unless and until the person or
persons requesting the issuance thereof shall have paid to the Corporation the
amount of any such tax or shall have established to the satisfaction of the
Corporation that such tax has been paid.

     (c)  CERTAIN DEFINITIONS. As used herein, the following terms shall have
the meanings set forth below:

     "Closing Price" of any security on any date shall mean the closing price of
such security on such date on the principal securities exchange or market on
which such security is traded.

     "Conversion Amount" shall mean an amount equal to Three Dollars ($3.00).

     "Conversion Date" shall mean the date on which the notice of call is given
by the Corporation.

     (d)  OTHER PROVISIONS. Notwithstanding anything in this Section 9, no
change in the Conversion Amount shall be made that would result in a Conversion
Price of less than the par value of the Common Stock.

     The Corporation (and any successor corporation) shall take all action
necessary so that a number of shares of the authorized but unissued Common Stock
(or common stock in the case of any successor corporation) sufficient to provide
for the conversion of the Series B Preferred Stock outstanding upon the basis
hereinbefore provided is at all times reserved by the Corporation (or any
successor corporation), free from preemptive rights, for such conversion,
subject to the provisions of the next succeeding paragraph. If the Corporation
shall issue any securities or make any change in its capital structure which
would change the number of shares of Common Stock into which each share of the
Series B Preferred Stock shall be convertible as herein provided, the
Corporation shall at the same time also make proper provision so that thereafter
there shall be a sufficient number of shares of Common Stock authorized and
reserved, free from preemptive rights, for conversion of the outstanding Series
B Preferred Stock on the new basis. If at any time the number of authorized but
unissued shares of Common Stock shall not be sufficient to effect the conversion
of all of the outstanding shares of Series B Preferred Stock, the Corporation
promptly shall seek such corporate action as may, in the opinion of its counsel,
be necessary to increase its authorized but unissued shares of Common Stock to
such number of shares as shall be sufficient for such purpose.

     In case of any consolidation with, or merger of the Corporation into, any
other entity, or any merger of another entity into the Corporation (other than a
merger which does not result in any reclassification, conversion, exchange or
cancellation of outstanding shares of Common Stock of the Corporation), or in
case of any sale or transfer of all or a majority of the assets of the
Corporation, or in the case of any share exchange pursuant to which all of the
outstanding shares of Common Stock are converted into other securities or
property, the Corporation shall make appropriate provision or cause appropriate
provision to be made so that each holder of shares of Series B Preferred Stock
then outstanding shall have the right thereafter to convert such shares of
Series B Preferred Stock into the kind and amount of shares of stock and other
securities and property receivable upon such consolidation, merger, sale,
transfer or share exchange by a holder of the number of shares of Common Stock
into which such shares of Series B Preferred Stock could have been converted
immediately prior to the effective date of such consolidation, merger, sale,
transfer or share exchange. In such event, the Conversion Price shall be the
lowest of (a) the amount determined pursuant to Section 9(a), (b) the average
Closing Price of the Common Stock for the five trading days ending on the tenth
trading day preceding the first public announcement of the pricing terms of the
consolidation, merger or other transaction or (c) the average Closing Price of
the Common Stock for the five trading days ending on the tenth trading day prior
to the first public announcement, if any, that the Corporation was considering a
sale or other disposition of its business or was engaged in acquisition
discussions or had received an acquisition offer. In the event of a Tender
Offer, the Conversion Price shall be the lowest of (i) the amount determined
pursuant Section 9(a), (ii) the average Closing Price of the Common Stock for
the five trading days ending on the tenth trading day preceding the first public
announcement of the pricing terms of the Tender Offer or (iii) the average
Closing Price of the Common Stock for the five trading days ending on the tenth
trading day prior to the first public announcement, if any, that the Corporation
was considering a sale or other disposition of its business or was engaged in
acquisition discussions or had received an acquisition offer; in such event, the
holders of the Series B Preferred Stock shall be entitled to convert such shares
into Common Stock on the condition that such shares of Common Stock are
purchased in the Tender Offer. If, in connection with any such consolidation,
merger, sale, transfer, or share exchange, each holder of shares of Common Stock
is entitled to elect to receive either securities, cash, or other assets upon
completion of such transaction, the Corporation shall provide or cause to be
provided to each holder of Series B Preferred Stock an equivalent right on the
same terms and subject to the same conditions applicable to holders of Common
Stock (including, without limitation, notice of the right to elect, limitations
on the period in which such election shall be made, and the effect of failing to
exercise the election). The Corporation shall not effect any such transaction
unless the


<PAGE>   4

provisions of this paragraph have been complied with. The above provisions shall
similarly apply to successive consolidations, mergers, sales, transfers, share
exchanges or Tender Offers.

     No fractional shares of Common Stock shall be issued upon conversion of
Series B Preferred Stock but, in lieu of any fraction of a share of Common Stock
which would otherwise be issuable in respect of the aggregate number of such
shares surrendered for conversion at one time by the same holder, the
Corporation at its option (a) may pay in cash an amount equal to the product of
(i) the arithmetic average of the Closing Price of a share of Common Stock on
the five consecutive trading days ending three trading days prior to the
Conversion Date and (ii) such fraction of a share or (b) may issue an additional
share of Common Stock.

     In case the Corporation shall issue rights or warrants on a pro rata basis
to all holders of the Common Stock entitling such holders to subscribe for or
purchase Common Stock or other securities, or if the Corporation shall, by
dividend or otherwise, distribute to all holders of its Common Stock evidences
of its indebtedness or assets, the holders of Series B Preferred Stock shall be
entitled to participate in the transaction or distribution on a basis and with
notice that the Board of Directors determines to be fair to the holders of the
Series B Preferred Stock and appropriate in light of the basis on which holders
of the Common Stock are to participate in the transaction.

     Whenever the Corporation shall propose to take any of the actions specified
in the third paragraph of this Section 9(d), the Corporation shall cause a
notice to be mailed at least twenty (20) days prior to the date on which the
books of the Corporation will close or on which a record will be taken for such
action, to the holders of record of the outstanding Series B Preferred Stock on
the third business day prior to the date of such notice. Such notice shall
specify the action proposed to be taken by the Corporation and the date as of
which holders of record of the Common Stock shall participate in any such
actions or be entitled to exchange their Common Stock for securities or other
property, as the case may be. Failure by the Corporation to mail the notice or
any defect in such notice shall not affect the validity of the transaction.

     (e)  MANDATORY CONVERSION. On August 3, 2002, (the "Mandatory Conversion
Date") all of the shares of Series B Preferred Stock then outstanding shall be
automatically converted, in accordance with the provisions of Section 9(a)-(d)
into shares of Common Stock as if the Mandatory Conversion Date were the
Conversion Date as to all such shares.

     SECTION 10. VOTING RIGHTS. Except as otherwise required by law or expressly
provided herein, shares of Series B Preferred Stock shall not be entitled to
vote on any matter.

     The affirmative vote or consent of the holders of a majority of the
outstanding shares of the Series B Preferred Stock, voting separately as a
class, will be required for (1) any amendment, alteration, or repeal, whether by
merger or consolidation or otherwise, of the Corporation's Certificate of
Incorporation if the amendment, alteration, or repeal materially and adversely
affects the powers, preferences, or special rights of the Series B Preferred
Stock, or (2) the creation and issuance of any Senior Liquidation Stock;
provided, however, that any increase in the authorized Preferred Stock of the
Corporation or the creation and issuance of any stock which is Junior Stock or
any other capital stock of the Corporation ranking on a parity with the Series B
Preferred Stock shall not be deemed to affect materially and adversely such
powers, preferences, or special rights.

     SECTION 11. OUTSTANDING SHARES. For purposes of this Certificate of
Designations, the following shares of Series B Preferred Stock shall not be
deemed outstanding: (i) all shares of Series B Preferred Stock from and after
the Conversion Date and (ii) all shares of Series B Preferred Stock held of
record by the Corporation or any subsidiary of the Corporation from the date of
registration of transfer to the Corporation or such subsidiary.

                      *** [NEXT PAGE IS SIGNATURE PAGE] ***

<PAGE>   5

     IN WITNESS WHEREOF, International Remote Imaging Systems, Inc. has caused
this Certificate of Designations to be executed by its Vice President, Finance
and Administration as of the 30th day of July 1999.


                                        By: /s/ Martin S. McDermut
                                            ------------------------------------
                                            Martin S. McDermut
                                            Vice President, Finance and
                                            Administration, Chief Financial
                                            Officer and Secretary


                                       27