EXHIBIT 4.10
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                          LIQUIDATED DAMAGES AGREEMENT

     THIS LIQUIDATED DAMAGES AGREEMENT (the "Agreement") is made and entered
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into as of August 7, 1998 by and among GREATER BAY BANCORP, a California
corporation (the "Company"), GBB CAPITAL II, a business trust formed under the
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laws of the state of Delaware (the "Trust") and SANDLER O'NEILL & PARTNERS, L.P.
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("Sandler O'Neill" or the "Initial Purchaser").
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     WHEREAS, as an inducement to the Initial Purchaser to enter into the
Purchase Agreement, dated August 7, 1998  (the "Purchase Agreement"), by and
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among the Company, the Trust and the Initial Purchaser (providing for, among
other things, the sale by the Trust to the Initial Purchaser of 30,000 of the
Trust's Floating Rate Capital Securities, Series A, liquidation amount of $1,000
per Capital Security (the "Capital Securities"), the proceeds of which will be
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used by the Trust to purchase Floating Rate Junior Subordinated Deferrable
Interest Debentures due September 15, 2028, Series A, of the Company (the
                                                                         
"Subordinated Debentures")), and as a condition to the several obligations of
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the Initial Purchaser thereunder, the Company and the Trust have agreed to
provide to the Initial Purchaser and its direct and indirect transferees certain
registration and related rights pursuant to and in accordance with the terms of
the Registration Rights Agreement, dated the date hereof (the "Registration
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Rights Agreement"), by and among the Company, the Trust and the Initial
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Purchaser; and

     WHEREAS, notwithstanding the fact that the Company and the Trust have
consummated or will consummate an Exchange Offer, pursuant to Section 2(b) of
the Registration Rights Agreement, the Initial Purchaser may, under certain
circumstances, require the Company and the Trust to file a Shelf Registration
Statement for the resale of certain Registrable Securities held by it;

     WHEREAS, the Registration Rights Agreement contains certain provisions
concerning the time within which the Company and the Trust must file the Shelf
Registration Statement and the period for which such Shelf Registration
Statement must remain effective and usable for resales; and

     WHEREAS, the Company, the Trust and the Initial Purchaser desire to provide
for the payment of liquidated damages by the Company directly to the Initial
Purchaser in the event that the Company and the Trust fail to comply with such
contractual provisions, as more fully set forth herein.

     NOW, THEREFORE, in consideration of the foregoing, and for other good and
valuable consideration the receipt of which is hereby acknowledged, the parties
hereto agree as follows:

     1.     Definitions. Capitalized terms used (including in the foregoing
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recitals) but not defined herein shall have the meanings given to such terms in
the Registration Rights Agreement, except that (a) the term "Shelf Registration
Statement" shall refer only to a Shelf Registration Statement filed by the
Company and the Trust pursuant to Section 2(b) of the Registration Rights

 
Agreement, and (b) the term "Registrable Securities" shall refer only to those
Registrable Securities held at such time by the Initial Purchaser.

     2.   Payment of Liquidated Damages. (a) In the event that (i) the Shelf
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Registration Statement is not filed with the SEC on or prior to the 45/th/ day
after a request for such filing is properly made by the Initial Purchaser in
accordance with Section 2(b) of the Registration Rights Agreement (provided that
in no event shall such date be required to be earlier than 75 days after the
Issue Date), or (ii) the Shelf Registration Statement is not declared effective
by the SEC on or prior to the later of the 40th day after the date such Shelf
Registration Statement was required to be filed pursuant to the terms of the
Registration Rights Agreement and the 180th date after the Issue Date, or (iii)
the Shelf Registration Statement has been declared effective and such Shelf
Registration Statement ceases to be continuously effective or usable for resales
(whether as a result of an event contemplated by Section 3(e) of the
Registration Rights Agreement or otherwise) at any time during the 180-day
period (and any extensions of such period pursuant to the last paragraph of
Section 3 of the Registration Rights Agreement) immediately following the date
on which the Shelf Registration Statement is first declared effective (other
than after such time as all Registrable Securities have been disposed of
thereunder or otherwise cease to be Registrable Securities pursuant to the terms
of the Registration Rights Agreement), then in each case the Company shall pay
liquidated damages to the Initial Purchaser, at a rate of 25 basis points per
annum in respect of the aggregate liquidation amount of Capital Securities held
by the Initial Purchaser or, in the event that the Trust is liquidated and
Subordinated Debentures are distributed to holders of Capital Securities, the
aggregate principal amount of Subordinated Debentures held by the Initial
Purchaser, as the case may be, in respect of the period (x) commencing on the
46th day after such request for the filing of a Shelf Registration Statement is
made by the Initial Purchaser (provided that in no event shall such date be
required to be earlier than 76 days after the Issue Date) and terminating upon
the filing of the Shelf Registration Statement (in the case of clause (i)
above), (y) commencing on the later of the 41st day after the date the Shelf
Registration Statement was required to be filed and the 181st day after the
Issue Date and terminating upon the effectiveness of the Shelf Registration
Statement (in the case of clause (ii) above), or (z) commencing on the day the
Shelf Registration Statement ceases to be effective or usable for resales and
terminating at such time as the Shelf Registration Statement again becomes
effective and usable for resales (in the case of clause (iii) above), provided,
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however, that the maximum aggregate amount of liquidated damages payable by the
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Company pursuant to this Agreement and the Registration Rights Agreement shall
not exceed 25 basis points per annum in respect of the aggregate liquidation
amount of the Capital Securities or, in the event the Trust is liquidated and
Subordinated Debentures are distributed to holders of Capital Securities, the
aggregate principal amount of Subordinated Debentures.

     (b) Any amounts of liquidated damages payable by the Company pursuant to
this Section 2 shall be paid in cash directly to the Initial Purchaser on the
next succeeding March 15, June 15, September 15 or December 15, as the case may
be, following the period in respect of which such Liquidated Damages have become
due and payable hereunder.

     3.   General.
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                    (1)  Counterparts. This Agreement may be executed in any
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                         number of counterparts and by the parties hereto in
                         separate counterparts, each of which when so executed
                         shall be deemed to be an original and all of which
                         taken together shall constitute one and the same
                         agreement.

                    (2)  Amendments. This Agreement may be amended by the
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                         parties hereto by a written instrument duly executed on
                         behalf of each of the parties hereto.

                    (3)  Entire Agreement. This Agreement and the Registration
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                         Rights Agreement constitute the entire agreement, and
                         supersede all prior agreements and understandings, both
                         written and oral, among the parties with respect to the
                         subject matter hereof.

                    (4)  Governing Law. This Agreement shall be governed by and
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                         construed in accordance with the laws of the State of
                         New York, without regard to any applicable conflicts of
                         law principles.

                    (5)  Notices.  All notices and other communications
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                         hereunder shall be in writing and shall be deemed given
                         if delivered to the parties at the addresses set forth
                         in, and in a manner contemplated by, the Registration
                         Rights Agreement.

                    (6)  Effective Date.  This Agreement shall be of no force
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                         and effect and the parties shall have no obligations or
                         rights hereunder until the Closing Time whereupon this
                         Agreement shall be effective.

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          IN WITNESS WHEREOF, the parties have executed this Liquidated Damages
Agreement as of the date first written above.

                              GREATER BAY BANCORP


                              By:    /s/ Steven C. Smith
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                              Name:  Steven C. Smith
                              Title: Executive Vice President, Chief Operating
                                     Officer and Chief Financial Officer


                              GBB CAPITAL II

 
                              By:    Greater Bay Bancorp,
                                     as Sponsor
 
                              By:    /s/ Steven C. Smith
                                     -------------------------------------------
                              Name:  Steven C. Smith
                              Title: Executive Vice President, Chief Operating 
                                     Officer and Chief Financial Officer


                              SANDLER O'NEILL & PARTNERS, L.P.

                              By:    Sandler O'Neill & Partners Corp.,
                                     the sole general partner


                              By:    /s/ Catherine A. Lawton
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                                     Catherine A. Lawton
                                     Vice President

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