SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20552

                                   ----------
(MARK ONE)
[X] QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934

    For the quarterly period ended June 30, 1998


[ ] TRANSITION REPORT PURSUANT TO SECTION 13 or 15(d) OF THE SECURITIES EXCHANGE
    ACT OF 1934

    For the transition period from___________________ to ___________________


                           COMMISSION FILE NO. 0-22499

                            BAYONNE BANCSHARES, INC.
             ------------------------------------------------------
             (Exact name of registrant as specified in its charter)

         Delaware                                            22-3511899
-------------------------------                          --------------------
(State or other jurisdiction of                           (I.R.S. Employer
incorporation or organization)                          Identification Number)

                     568 Broadway, Bayonne, New Jersey 07002
                    ----------------------------------------
                    (Address of principal executive offices)

       Registrant's telephone number, including area code: (201) 437-1000

--------------------------------------------------------------------------------

Former name, former address and former fiscal year, if changed since last report

     Indicate by check |X| whether the Registrant (1) has filed all reports
required to be filed by Section 13 or 15 (d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
Registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.

                                 Yes |X| No [ ]

            AS OF AUGUST 14, 1998, THERE WERE 9,094,495 SHARES OF THE
                      REGISTRANT'S COMMON STOCK OUTSTANDING



<PAGE>



                            BAYONNE BANCSHARES, INC.

                                      INDEX

PART I. FINANCIAL INFORMATION                                              Page
                                                                           ----
Item 1.  Consolidated Financial Statements

         Consolidated Statements of Financial Condition as of                
         June 30, 1998, and March 31, 1998 ..............................    1

         Consolidated Statements of  Income for the three                    
           months ended June 30, 1998 and 1997 ..........................    2

         Consolidated Statements of Stockholders' Equity for the three       
           months ended June 30, 1998 and 1997 ..........................    3

         Consolidated Statements of Cash Flows for the three                 
           months ended June 30, 1998 and 1997 ..........................  4-5

         Notes to Consolidated Financial Statements .....................  6-7

Item 2.  Management's Discussion and Analysis of Financial                    
         Condition and Results of Operations ............................ 8-11

Item 3.  Quantitative and Qualitative Disclosures About Market Risk .....   11

PART II. OTHER INFORMATION ..............................................   12



<PAGE>
PART I                       FINANCIAL INFORMATION

Item 1                 Consolidated Financial Statements

                            BAYONNE BANCSHARES, INC.
                 CONSOLIDATED STATEMENTS OF FINANCIAL CONDITION
                           (In Thousands Of Dollars)

<TABLE>
<CAPTION>

             ASSETS                                             June 30, 1998         March 31, 1998
             ------                                             -------------         --------------
                                                                 (unaudited)
<S>                                                                 <C>                  <C>    
Cash and cash equivalents:
  Cash on hand and in banks                                         $ 5,198              $ 4,712
  Deposits with financial institutions                                  850               11,900
  Short-term investments                                                  5                    5
                                                                   --------            ---------
    Total cash and cash equivalents                                   6,053               16,617

Securities available for sale, at market value                      339,228              362,778
Securities held to maturity                                          12,169               10,274
Loans receivable, net                                               318,096              235,465
Accrued interest receivable, net                                      4,300                4,089
Federal Home Loan Bank stock, at cost                                10,511                7,460
Real estate acquired in settlement of loans                             266                  330
Office properties and equipment, net                                  5,463                5,351
Prepaid expenses                                                        648                  672
Other assets                                                          3,491                3,022
                                                                  ---------            ---------
 Total Assets                                                     $ 700,225            $ 646,058
                                                                  =========            =========

        LIABILITIES AND STOCKHOLDERS' EQUITY

Liabilities:
 Deposits                                                         $ 421,728            $ 423,545
 Borrowings                                                         174,226              115,000
 ESOP debt                                                              326                  380
 Advance payments by borrowers for taxes and insurance                2,211                3,754
 Accrued expenses and other liabilities                               5,839                4,730
                                                                  ---------            ---------
  Total Liabilities                                               $ 604,330            $ 547,409
                                                                  =========            =========

Stockholders' equity:
 Common stock                                                     $      91            $      91
 Additional paid in capital                                          60,576               60,246
 Common stock acquired by ESOP and MRP                               (7,320)              (4,301)
 Retained earnings-substantially restricted                          44,248               43,702
 Comprehensive income (loss)                                         (1,700)              (1,089)
                                                                  ---------            --------- 
  Total stockholders' equity                                         95,895               98,649
                                                                  ---------            ---------
   Total liabilities and stockholders' equity                     $ 700,225            $ 646,058
                                                                  =========            =========
</TABLE>

        See accompanying notes to the consolidated financial statements.

                                       1
<PAGE>


                            BAYONNE BANCSHARES, INC.
                                AND SUBSIDIARIES
                        CONSOLIDATED STATEMENTS OF INCOME
                           (In Thousands Of Dollars)
                                   (Unaudited)





                                                         Three Months Ended
                                                             June  30
                                                      ------------------------
                                                         1998           1997
                                                      ---------        -------
Interest and dividend income:
  Loans                                                $ 4,919         $4,916
  Securities available for sale                          5,347          4,584
  Securities held to maturity                              323            305
  Overnight deposits with financial institutions            58             87
                                                       -------         ------
Total interest and dividend income                      10,647          9,892
                                                       -------         ------

Interest expense:
  Interest on deposits                                   4,592          5,186
  Interest on other borrowings                           1,815          1,129
                                                       -------         ------
Total interest expense                                   6,407          6,315
                                                       -------         ------
Net interest income                                      4,240          3,577
  Provision for loan losses                                 50             45
                                                       -------         ------
Net interest income after  provision for loan losses     4,190          3,532
                                                       -------         ------

Other income:
 Loan fees and service charges                              68             68
 Deposit  fees                                             159            131
 Loss on securities available for sale, net                (21)             -
 Income from Bayonne Service Corp.                          46             47
  Gain (loss) on sales of real estate acquired in
   settlement of loans, net                                (14)            12
  Other                                                     73            140
                                                       -------         ------
Total other income                                         311            398
                                                       -------         ------

Operating expenses:
  Compensation and employee benefits                     1,575          1,466
  Occupancy and equipment                                  310            291
  Data processing service expense                          211            224
  Deposit insurance premiums                               137            145
  Other                                                    500            474
                                                       -------         ------
Total operating expenses                                 2,733          2,600
                                                       -------         ------


Income before income tax expense                         1,768          1,330
Income tax expense                                         654            492
                                                       -------         ------
Net income                                             $ 1,114         $  838
                                                       =======         ======
Basic earnings per share                               $  0.12         $ 0.09
                                                       =======         ======
Diluted earnings per share                             $  0.12         $ 0.09
                                                       =======         ======

       See accompanying notes to the consolidated financial statements.

                                       2

<PAGE>


                            BAYONNE BANCSHARES, INC.
                                AND SUBSIDIARIES
                 CONSOLIDATED STATEMENTS OF STOCKHOLDERS' EQUITY
                THREE MONTH PERIODS ENDED JUNE 30, 1998 AND 1997
                           (In Thousands Of Dollars)
                                   (Unaudited)

<TABLE>
<CAPTION>

                                                                                      Retained     
                                              Additional    Unallocated               Earnings                           Total
                                 Common        Paid in        ESOP         MRP      Substantially   Comprehensive     Stockholders'
                                 Stock         Capital       Shares       Shares     Restricted      income (loss)       Equity
----------------------------------------------------------------------------------------------------------------------------------
<S>                              <C>           <C>         <C>          <C>          <C>                <C>              <C>

 Balance at March 31, 1997        $90          $12,643     $  (598)     $  (378)      $ 40,658          $(4,336)         $48,079
 Net income for the three                                                                            
        month period ended                                                                           
        June 30, 1997                                                                      838                               838
 Amortization of ESOP                               80          55                                                           135
 Amortization of MRP                                26                       28                                               54
 Dividends paid                                                                          (176)                              (176)
 Comprehensive income (loss)       -                 -           -            -             -             1,750            1,750
                                 ---           -------     -------      -------       --------          -------          -------
 Balance at                                                                                          
    'June 30, 1997               $90           $12,749     $  (543)     $  (350)      $ 41,320          $(2,586)         $50,680
                                 ---           -------     -------      -------       --------          -------          -------
                                                                                                     
                                                                                                     
 Balance at March 31, 1998       $91           $60,246     $(4,081)     $  (220)      $ 43,702          $(1,089)         $98,649
 Net Income for the three                                                                            
        month period ended                                                                           
        June 30, 1998                                -                                   1,114                             1,114
 Amortization of ESOP                              254         112                                                           366
 Amortization of MRP                                72                      124                                              196
 Dividends paid                                                                           (568)                             (568)
 Exercise of stock options                           4                                                                         4
 Purchase of shares for
  MRP Plan                                                               (3,255)                                          (3,255)
 Comprehensive income (loss)       -                 -           -            -              -             (611)            (611)
                                 ---           -------     -------      -------       --------          -------          -------
 Balance at                                                                                          
  'June 30, 1998                 $91           $60,576     $(3,969)     $(3,351)      $ 44,248          $(1,700)         $95,895
                                 ---           -------     -------      -------       --------          -------          -------
</TABLE>


       See accompanying notes to the consolidated financial statements.


                                       3


<PAGE>



                            BAYONNE BANCSHARES, INC.
                                AND SUBSIDIARIES
                      CONSOLIDATED STATEMENTS OF CASH FLOWS
                           (In Thousands Of Dollars)
                                   (Unaudited)


<TABLE>
<CAPTION>


                                                                                        Three months ended June 30
                                                                                    --------------------------------
                                                                                     1998                      1997
                                                                                    -------                   -------
 <S>                                                                                <C>                        <C> 

 Cash flows from operating activities:
  Net income                                                                        $ 1,114                    $ 838

Adjustments to reconcile net income to net cash
 provided by (used in) operating activities:

  Depreciation of office properties and equipment                                       112                      146
  Loss on securities available for sale                                                  21                        -
  Provision for loan losses                                                              50                       45
  Loss on real estate operations                                                        (14)                      12
  Amortization (accretion) of premiums and discount on
   investment securities and mortgage backed securities                                 669                       73
  Amortization of ESOP and MRP shares                                                   562                      189
  Net decrease in deferred loan fees                                                    (73)                     (66)
  Net increase in accrued interest receivable                                          (211)                    (190)
  Net decrease (increase) in prepaid expenses                                            24                     (302)
  Net (increase) decrease in other assets                                              (406)                     274
  Net increase (decrease) in accrued expenses and other liabilities                   1,109                   (1,611)
                                                                                    --------                 --------
     Total adjustments                                                                1,843                   (1,430)
                                                                                    --------                 --------

    Net cash provided by (used in) operating activities                               2,957                     (592)
                                                                                    --------                 --------

Cash flows from investing activities:

  Principal payments and maturities on securities
   available for sale                                                                43,081                    9,430
  Proceeds from sale of securities available for sale                                62,076                        -
  Purchases of securities available for sale                                        (84,840)                 (46,390)
  Net (increase) decrease in loans receivable                                       (82,681)                   2,577
  Maturities of securities held to maturity                                             100                       49
  Proceeds from sales of real estate acquired in
  settlement of loans                                                                    25                        -
  Purchase of FHLB stock                                                             (3,051)                       -
  Purchases of premises and equipment                                                  (224)                     (25)
                                                                                    --------                 --------
     Net cash used in investing activities                                          (65,514)                 (34,359)
                                                                                    --------                 --------

  Cash flows from financing activities:
  Net decrease in deposits                                                           (1,817)                  (1,350)
  Net (decrease) increase in advance payments by borrowers for
   taxes and insurance                                                               (1,543)                     112
  Dividends paid                                                                       (568)                    (176)
  Increase in borrowings and repurchase agreements                                   59,226                   25,500
  Payment on ESOP debt                                                                  (54)                     (55)
  Purchase of shares for MRP Plan                                                    (3,255)                       -
  Exercise of stock options                                                               4                        -
                                                                                    --------                 --------
    Net cash  provided  by financing activities                                      51,993                   24,031
                                                                                    --------                 --------
 
</TABLE>


       See accompanying notes to the consolidated financial statements.

                                        4

<PAGE>



                            BAYONNE BANCSHARES, INC.
                                AND SUBSIDIARIES
                      CONSOLIDATED STATEMENTS OF CASH FLOWS
                                   (Continued)
                           (In Thousands Of Dollars)
                                   (Unaudited)

<TABLE>
<CAPTION>

                                                                     Three Months Ended June 30
                                                                   ------------------------------
                                                                      1998                 1997
                                                                   ----------            --------
<S>                                                                <C>                   <C>

Net decrease in cash and cash equivalents                          (10,564)              (10,920)

Cash and cash equivalents at start of period                        16,617                15,472
                                                                   --------              --------

Cash and cash equivalents at end of period                         $ 6,053               $ 4,552
                                                                   ========              +=======

Supplemental schedule of cash flow information:

Real estate acquired in settlement of loans                        $     -               $   158
                                                                   --------              --------
Taxes Paid                                                         $   956               $     -
                                                                   --------              --------
Interest Paid                                                      $ 6,391               $ 6,938
                                                                   --------              --------

</TABLE>



       See accompanying notes to the consolidated financial statements.

                                        5



<PAGE>




                    BAYONNE BANCSHARES, INC. AND SUBSIDIARIES
                   NOTES TO CONSOLIDATED FINANCIAL STATEMENTS

1) BASIS OF PRESENTATION

     The accompanying unaudited consolidated financial statements have been
prepared in accordance with generally accepted accounting principles for interim
financial information and in conformity with the instructions to Form 10-Q and
Article 10 of Regulation S-X for Bayonne Bancshares, Inc. and Subsidiaries (the
Company, the Holding Company of First Savings Bank of New Jersey, SLA.)


2) PLAN OF CONVERSION

     On August 22, 1997, Bayonne Bancshares, Inc. in accordance with the Bank's
plan of conversion and reorganization sold 4,869,190 shares of common stock at a
price per share of $10.00. In addition, each of the 1,410,735 shares of common
stock of First Savings Bank of New Jersey, SLA held by public stockholders were
exchanged for 2.933 shares of common stock of Bayonne Bancshares, Inc. in
accordance with the Bank's Plan of Conversion and Agreement and Plan of
Reorganization.

3) NON-PERFORMING LOANS

     Non-performing loans at June 30, 1998, and March 31, 1998 are as follows
(in thousands of dollars):

                                            June 30, 1998       March 31, 1998
                                            -------------       --------------
Loans delinquent 90 days or more and
 other non-performing loans                     $2,792              $3,817
                                                ------              ------

Loans delinquent 90 days or more and
 other non-performing loans as a
 percentage of  loans receivable, net             0.88%               1.62%

     An analysis of the allowance for loan losses for the three-month periods
ended June 30, 1998 and 1997 is as follows (in thousands of dollars)

                                          June 30, 1998          June 30, 1997
                                          -------------          -------------
Balance at start of period                    $2,953                 $3,158
Provision charged to operations                   50                     45
Net (charge-offs) recoveries                     (52)                    --
                                              ------                 ------
Balance at end of period                      $2,951                 $3,203
                                              ======                 ======
                                       6
<PAGE>

4) EARNINGS PER SHARE

     The Bank adopted SFAS No. 128, "Earnings Per Share," on December 31, 1997.
SFAS No. 128 established the new standard for computation and presentation of
net income per common share. Under the new requirements both basic and diluted
net income per common share is presented. All prior period net income per common
share data has been restated.

     Basic net income per common share is calculated by dividing net income,
less the dividends on preferred stock, if any, by the average common shares
outstanding during the period.

     Diluted net income per common share is calculated similar to that of the
basic net income per common share except that the denominator is increased to
include the number of additional common shares that would have been outstanding
if all potentially dilutive common shares arising from unexercised stock options
were issued during the reporting period.

     As discussed in note 2, the Company completed a conversion and
reorganization, which included the exchange of previously outstanding shares for
new shares of the stock holding company, Bayonne Bancshares, Inc., at an
exchange ratio of 2.933 to 1. All historical share and per share information has
been adjusted to reflect this change unless otherwise noted.

     The following table summarizes the computation of basis earnings and
diluted earnings per common share for the three months ended June 30, 1998 and
1997:

                                                          Three months ended
                                                               June 30
                                                       -------------------------
(In thousands, except per share data)                     1998           1997
                                                       ----------     ----------
Earnings available to common shareholders              $    1,114     $      838
Weighted average common shares outstanding              8,941,610      8,824,010
Plus common stock equivalents                              34,773         94,029
Diluted weighted average shares outstanding             8,976,343      8,918,039
Earnings per common share:                         
  Basic                                                      0.12           0.09
  Diluted                                                    0.12           0.09
                                                 

5) RECLASSIFCATIONS

     Certain reclassifications have been made to the 1997 amounts to conform to
the 1998 presentation.


6) PROPOSED MERGER WITH RICHMOND COUNTY FINANCIAL CORP.

     The Company announced on July 20, 1998 that it had entered into a
definitive agreement to merge with Richmond County Financial Corp. Under the
terms of the agreement, which is subject to approval by shareholders of each
company and regulatory authorities, the Company's shareholders will receive 1.05
shares of Richmond County Financial Corp. for each share of the Company owned.
The transaction is intended to be a tax free merger accounted for as a pooling
of interests and is expected to be completed by the first quarter of calendar
1999. The Agreement and Plan of Merger was included as an exhibit to the Form
8-K filed by the Company on July 27, 1998.


7) RECENT ACCOUNTING PRONOUNCEMENT

     In June 1998, the Financial Accounting Standards Board issued SFAS No. 133,
"Accounting for Derivative Instruments and Hedging Activities." This statement
establishes accounting and reporting standards for derivative instruments, and
for hedging activities. This statement is effective for periods commencing
January 1, 2000. The Corporation is currently evaluating the impact this
statement will have on its results of operations and its financial position. The
effect of adopting this statement is not expected to have a material effect on
the Company's results of operations or financial condition.


8) COMPREHENSIVE INCOME

     During the quarter ended June 30, 1998, the Company adopted SFAS No. 130,
"Reporting Comprehensive Income." SFAS No. 130 requires the reporting of
comprehensive income in addition to net income from operations. Comprehensive
income is a more inclusive financial reporting methodology that includes
disclosure of certain financial information that historically has not been
recognized in the calculation of net income. The financial statements reflect
the adoption of SFAS No. 130.

     Total comprehensive income (loss) for the three months ended June 30, 1998
and 1997 was $503,000 and 2,588,000, respectively. For the three months ended
June 30, 1998, total comprehensive income (loss) represented net income of
$1,114,000 and other comprehensive income (loss) of ($611,000). For the three
months ended June 30, 1997, total comprehensive income (loss) represented net
income of $838,000 and other comprehensive income (loss) of $1,750,000. Other
comprehensive income for the above-mentioned periods related to unrealized
gains/losses on securities available for sale, net of tax.


                                       7
<PAGE>




ITEM 2. MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND 
RESULTS OF OPERATIONS

     FINANCIAL CONDITION

     Stockholders' equity decreased by $2.7 million to $95.9 million, at June
30, 1998 from $98.6 million at March 31, 1998. The decrease in stockholders'
equity was primarily attributable to the acquisition of 194,768 shares of
common stock for approximately $3.2 million during the quarter to be used for
stock awards under the Company's 1998 Management Recognition Plan. The decrease
was partially offset by net income of $1.1 million for the three months ended
June 30, 1998. At June 30, 1998, the Bank's tangible, core and risk based
capital ratios were 11.42%, 11.42%, and 25.96% respectively.

     Total assets increased by $54.1 million or 8.4% to $700.2 million at June
30, 1998 from $646.1 million at March 31, 1998. Loans receivable, net increase
by $82.6 million or 35.1 % to $318.1 million at June 30, 1998 from $235.5
million at March 31, 1998. On June 30, 1998, the Bank purchased an $80.0 million
mortgage loan package, which consisted of seasoned, fixed rate single family
loans yielding approximately 6.5%. Mortgage origination's during the quarter,
net of mortgage prepayments and amortization, totaled $2.7 million. Securities
available for sale, at market value decreased by $23.6 million or 6.5% to $339.2
million at June 30, 1998 from $362.8 million at March 31, 1998. The decrease was
primarily attributable to the sale of lower yielding U.S. Treasury, agency and
mortgage-backed securities totaling $57 million. The Bank also purchased $32.0
million of variable rate trust preferred securities and $35.0 million of
adjustable and fixed rate mortgages backed securities. Principal amortization on
mortgage backed securities during the quarter ended June 30, 1998 approximated
$42.9 million.

     Total liabilities increased by $56.9 million to $604.3 million at June 30,
1998 from $547.4 million at March 31, 1998. Total deposits decreased by $1.8
million to $421.7 million at June 30, 1998 from $423.5 million at March 31,
1998. This decrease was primarily due to maturing, long-term, high rate
certificates of deposit. Borrowings increased by $59.2 million to $174.2 million
at June 30, 1998 compared to $115.0 million at March 31, 1998. Borrowings were
used to increase the Company's holdings of adjustable rate trust preferred and
mortgage backed securities, to fund a portion of the bulk loan purchase of $80.0
million and to fund the outflow of maturing high yielding certificates of
deposit.

     RESULTS OF OPERATIONS

     Net income increased by $276,000 or 32.9% to $1.1 million for three months
ended June 30, 1998 from $838,000 for the three months ended June 30, 1997. Cash
earnings for the quarter increased stockholders' equity by $353,000 or 32% more
than the earnings reported above. Cash earnings represent the amount by which
stockholders' equity changes each period due to operating results, including
reported earnings and 


                                       8
<PAGE>


amortization related to certain employee stock plans and the related tax effect.
The increase resulted primarily from an increase in net interest income, which
was partially offset by an increase in total interest expense and total
operating expenses.

     Net interest income rose by 20.0% in the quarter ended June 30, 1998 to
$4.2 million compared to $3.6 million for the comparable prior year quarter. The
increase was the result of an increase in average interest earning assets of
$83.2 million or 14.7% to $648.9 million at June 30, 1998 from $565.6 million at
June 30, 1997 and an 8 basis point improvement in the net interest margin to
2.61% in the first quarter in fiscal 1999 compared to 2.53% in the first quarter
in fiscal 1998.

     Total interest expense increased $92,000 to $6.4 million for the three
months ended June 30, 1998 compared to $6.3 million for the three months ended
June 30, 1997. The increase was primarily attributable to the average balance of
borrowings increasing $56.1 million or 66.8% to $140.1 million compared to $84.0
million at June 30, 1997, offset by a decrease of $21.5 million or 4.8% in the
average balance of interest bearing savings deposits to $421.9 million at June
30, 1998 compared to $443.4 million for the comparable prior year.

     The provision for loan losses totaled $50,000 for the three months ended
June 30, 1998 as compared to $45,000 for the prior year's three month period.
The provision is a result of management's judgment based on a review of its loan
portfolio and consideration of a variety of factors including, 1) the risk
characteristics of its loan portfolio, 2) current economic conditions, 3) actual
losses previously experienced and 4) the existing levels of reserves for
possible loan losses in the future. Non-performing loans at June 30, 1998 and
March 31, 1998 were $2.8. million and $3.8 million or 0.88% and 1.62% of loans
receivable net, respectively.

     Non-interest income totaled $311,000 for the three months ended June 30,
1998 compared to $398,000 for quarter ended June 30, 1997. The 1997 fiscal
quarter included $95,000 in non-recurring income related to a previously
completed joint venture, while the three months ended June 30, 1998 included a
$21,000 loss on sale of available for sale securities.

     Operating expenses for the quarter ended June 30, 1998 increased by
$133,000 or 5.1% to $2.7 million, compared to $2.6 million for the quarter ended
June 30, 1997. Compensation and employee benefits rose by $109,000 or 7.4% to
$1.6 million for the three-months ended June 30, 1998 compared to $1.5 million
for the comparable prior year. Expenses for new and existing ESOP and management
recognition plans increased by $373,000 to $561,000 for the quarter ended June
30, 1998 compared to $188,000 for the quarter ended June 30, 1997, which were
partially offset by lower executive salary expense.

     Income tax expense for the quarter increased by $162,000 to $654,000 for
the three months ended June 30, 1998 from the $492,000 for the comparable prior
period, as a result of higher pre-tax income for the most recent three month
period.


                                       9
<PAGE>




CAPITAL

     The OTS requires that the Bank meet minimum tangible, core and risk-based
capital requirements. As of June 30, 1998, the Bank exceeded all regulatory
capital requirements. The Bank's required, actual and excess capital levels as
of June 30, 1998 were as follows (in thousands of dollars):
<TABLE>
<CAPTION>

                                                                               Excess of Actual
                             Required      % of         Actual       % of      Over Regulatory
                              Amount      Assets        Amount      Assets      Requirements
                             --------     ------       -------      ------     ---------------
<S>                          <C>           <C>         <C>          <C>           <C>    
Tangible Capital......       $10,190       1.50%       $77,547      11.42%        $67,357
Core Capital.........        $27,174       4.00%       $77,547      11.42%        $50,373
Risk-based Capital...        $23,893       8.00%       $77,547      25.96%        $53,654
</TABLE>

     If the allowance for marketable debt and equity securities was included in
capital, the Bank's tangible, core and risk-based capital ratios would be
11.22%, 11.22% and 25.53%, respectively, at June 30, 1998.

LIQUIDITY

     The Bank is required to maintain minimum levels of liquid assets as defined
by OTS regulations. This requirement, which varies from time to time depending
upon economic conditions and deposit flows, is based upon a percentage of
deposits and short-term borrowings. The required ratio currently is 5.0%. The
Bank's liquidity ratio averaged 39.83% during the month of June 30, 1998 and
35.03% during three months ended June 30, 1998. The Bank adjusts liquidity as
appropriate to meet its asset and liability management objectives.

YEAR 2000 CONSIDERATION

     The Company and the Bank relies on computers for the daily conduct of its
business and for data processing. There is concern among industry experts that
on January 1, 2000 computers will be unable to "read" the new year and there may
be widespread computer malfunctions. The Bank generally relies on independent
third parties to provide data processing services to the Bank, and has been
advised by its data processing service center that the issue is being addressed.
The Bank recognized that a comprehensive and coordinated plan of action was
needed to ensure readiness to perform Year 2000 processing. A Year 2000
Committee has been formed to implement the Year 2000 project, develop policies,
document readiness of the Bank and the Company to accommodate Year 2000
processing, and track and test progress towards full compliance. The Bank
contracts with service bureaus to provide the majority of its data processing
and is dependent upon purchased application software. In house applications are
limited to word-processing and spreadsheet functions.


                                       10
<PAGE>


     The Bank is in the process of ensuring that external vendors and services
are adequately addressing the system and software issues related to the Year
2000 by obtaining written system certifications that the systems are fully Year
2000 compliant or that the vendor has a plan to become fully compliant in the
very near future. The Bank will utilize the primary servicers end-to-end tests,
which will be available by December 31, 1999, to simulate daily processing on
sensitive century dates. In the evaluation, the Bank will ensure that critical
operations will continue if servicers or vendors are unable to achieve the Year
2000 requirements. The Committee completed a systems inventory and obtained
vendor certification. The Committee identified critical applications and
developed detailed plans for hardware/system upgrades and system replacements
where necessary. The Committee has budgeted $250,000 for hardware system
upgrades and system replacements which has been approved by the Board of
Directors. All upgrades have been implemented and will be substantially tested
by December 31, 1998.

ITEM 3. QUANTITATIVE AND QUALITATIVE DISCLOSURES ABOUT MARKET RISK

     There have been no material changes in information regarding quantitative
and qualitative disclosures about market risk as of June 30, 1998 from the
information as of March 31, 1998, which was disclosed in the Company's 1998 Form
10-K.


                                       11
<PAGE>



PART II. OTHER INFORMATION

Legal Proceedings

     In the normal course of business, the Company may be a party to various
outstanding legal proceedings and claims. In the opinion of management, the
financial position of the Company will not be materially affected by the outcome
of such legal proceedings and claims.

Changes in Securities
        Not applicable.

Defaults upon Senior Securities
        Not applicable.

Submission of Matters to Vote of Security Holders
        None.

Other Information

The Company announced on July 20, 1998 that it had entered into a definitive
agreement to merge with Richmond County Financial Corp. Under the terms of the
agreement, which is subject to approval by shareholders of each company and
regulatory authorities, the Company's shareholders will receive 1.05 shares of
Richmond County Financial Corp. for each share of the Company owned. The
transaction is intended to be a tax free merger accounted for as a pooling of
interests and is expected to be completed by the first quarter of calendar 1999.
The Agreement and Plan of Merger was included as an exhibit to the Form 8-K
filed by the Company on July 27, 1998.

Exhibits and Reports on Form 8-K

(a) The following exhibits are filed as part of this report:

            3.1 Certificate of Incorporation of Bayonne Bancshares, Inc.*
            3.2 Bylaws of Bayonne Bancshares, Inc.*
            4.0 Stock Certificate of Bayonne Bancshares, Inc.*
           11.0 Computation of earnings per share (filed herewith) 
           27.0 Financial Data Schedule (filed herewith)

(b) No reports on Form 8-K were filed for the quarter ended June 30, 1998.

* Incorporated herein by reference in this document from the Exhibits to the
  Form S-1 Registration Statement, filed on March 13, 1997, and any
  amendments thereto (Registration No. 333-23199).


                                       12


<PAGE>


                                   SIGNATURES

     Pursuant to the requirements of Section 13 or 15(d) of the Securities
Exchange Act of 1934, the Bank has duly caused this report to be signed on its
behalf by the undersigned, thereunto duly authorized.


                                            BAYONNE BANCSHARES, INC.


August 14, 1998                     By: /s/ MICHAEL NILAN
                                        ----------------------------------------
                                        Michael Nilan
                                        President and Chief Executive Officer


Date: August 14, 1998               By: /s/ EUGENE V. MALINOWSKI
                                        ----------------------------------------
                                        Eugene V. Malinowski
                                        Vice President and
                                          Chief Financial Officer


                                       13


<PAGE>


                                   EXHIBIT 11

                        Computation of Earnings Per Share



                               Three Months Ended June 30

                                            1998            1997
                                         ---------       ---------
                 
Net Income applicable                                 
To common stock and common                            
stock equivalents                                     
(in thousands)                               $1114          $838
                                             -----          ----
                                                      
Basic Earnings Per Share                              
Average shares of common                              
stock outstanding                        8,941,610       8,824,010
                                         ---------       ---------
                                                      
                                                      
Basic Earnings Per Share                     $0.12          $0.09
                                             -----          -----
                                                      
Fully Diluted Earnings                                
Per Share                                             
                                                      
                                                      
Average shares of common                              
stock outstanding - basic                8,941,610       8,824,010
                                                      
Common stock equivalents of                           
stock options and management                          
recognition plans                           34,733          94,029
                                         ---------       ---------
Average  shares of common                             
stock outstanding-diluted                8,976,343       8,918,039
                                         =========       =========
                                                      
                                                      
Fully Diluted Earnings                                
Per Share                                    $0.12          $0.09
                                             =====          =====
                                                   

NOTE:

The Company completed a conversion and reorganization, which included the
exchange of previously outstanding shares for new shares of the stock holding
company, Bayonne Bancshares, Inc. at an exchange ratio of 2.933 to 1. All
historical share and per share information has been adjusted to reflect this
change unless otherwise noted.


                                       14