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                                                                   EXHIBIT 10.51

                                 FIRST AMENDMENT
                                       TO
                       LIMITED LIABILITY COMPANY AGREEMENT

      This First Amendment (the "Amendment") to the Limited Liability Company
Agreement of Nalco LLC (the "Company"), dated as of May 17, 2004 (the
"Agreement"), is made and entered into as of December 30, 2005 by and among the
Sponsor Members named in the Agreement. Capitalized terms not otherwise defined
herein shall have the meanings ascribed to such terms in the Agreement.

      WHEREAS, pursuant to Section 3.14 of the Agreement, in the event that the
Company ceases to qualify as an "operating company" (as defined in 29 C.F.R. ss.
2510.3-101(c)), then the Company and each Member will cooperate in good faith to
take all reasonable action necessary to provide that the investment (or at least
51% of the investment valued at cost) of each Member that qualifies as a
"venture capital operating company" (as defined in 29 C.F.R. ss. 2510.3-101(d))
(a "VCOC Member") shall continue to qualify as a "venture capital investment"
(as defined in 29 C.F.R. ss. 2510.3-101(d));

      WHEREAS, upon the completion of the sale of shares of common stock of
Nalco Holding Company (the "Common Shares") by the Company on the date hereof,
the Company shall cease to qualify as an "operating company";

      WHEREAS, in connection with such event, the Company, pursuant to the
stockholders agreement dated as of November 16, 2004 by and among the Company,
the Sponsor Members and Nalco Holding Company, is obligated to distribute to
each VCOC Member, through a redemption of its Class A Units, the Common Shares
beneficially owned by such VCOC Member, as determined by the Board and the
pricing committee of the Board (such redemption, the "VCOC Redemption");

      WHEREAS, in connection with the VCOC Redemption and in accordance with
Section 3.14 of the Agreement, the parties desire to amend certain provisions of
the agreement to reflect the fact that the VCOC Members will hold Common Shares
directly and will no longer hold Class A Units; and

      WHEREAS, pursuant to Section 13.09 of thereof, the Agreement may be
amended by the agreement of two of the Investor Groups;

      NOW, THEREFORE, in consideration of the mutual covenants and agreements
contained herein and in the Agreement, and in consideration of other good and
valuable consideration, the receipt and sufficiency of which are hereby
acknowledged, the parties hereto, intending to be legally bound, hereby agree to
amend the Agreement as follows:

      1.    Ownership Thresholds. With respect to any provision of the Agreement
which provides for the determination of an Investor Group's beneficial ownership
percentage of Class

A Units relative to such Investor Group's Initial Equity Stake, such Investor
Group's number of "Class A Units" shall be deemed to include both (a) the number
of Class A Units that such Investor Group owns and (b)a number of Class A Units
having a value equivalent to that of the number of Common Shares owned by such
Investor Group (calculated based on the price per Class A Unit most recently
determined by the Board or the pricing committee of the Board).

      2.    Entire Agreement. The Agreement, this Amendment, the Management
Members Agreement and the Registration Rights Agreement set forth the entire
understanding and agreement of the parties hereto and supersede any and all
other understandings, term sheets, negotiations or agreements between the
parties hereto relating to the subject matter of this Agreement and the
Registration Rights Agreement. In the event of any conflict between a provision
of this Agreement and a provision of the Management Members Agreement, the
relevant provision of this Agreement, as amended, shall be deemed to control and
shall supersede the conflicting provision of the Management Members Agreement.

      3.    Governing Law. THIS AGREEMENT SHALL BE CONSTRUED IN ACCORDANCE WITH,
AND GOVERNED BY, THE LAWS OF THE STATE OF DELAWARE WITHOUT REGARD TO CONFLICTS
OF LAWS PRINCIPLES WHICH WOULD RESULT IN THE APPLICATION OF THE LAWS OF ANOTHER
JURISDICTION.

      4.    Severability. In the event that any one or more of the provisions
contained in this Agreement shall for any reason be held to be invalid, illegal
or unenforceable, the same shall not affect any other provision of this
Agreement, but this Agreement shall be construed in a manner which, as nearly as
possible, reflects the original intent of the parties.

      5.    Counterparts. This Agreement may be executed in counterparts, each
of which shall be deemed an original, and all of which together shall constitute
a single agreement.

      6.    Full Force and Effect of Agreement. The Agreement, as modified by
this Amendment, and as this Amendment is made a part thereof, shall remain in
full force and effect.

              [The remainder of this page intentionally left blank]

      IN WITNESS WHEREOF, the parties hereto have caused this Agreement to be
duly executed as of the date first written above.

                                    BLACKSTONE CAPITAL PARTNERS IV L.P.

                                        By: Blackstone Management Associates IV
                                            L.L.C., its General Partner

                                            By: /s/ Chinh Chu
                                                --------------------------------
                                                Name: Chinh Chu
                                                Title: Member

                                    BLACKSTONE CAPITAL PARTNERS IV-A L.P.

                                        By: Blackstone Management Associates IV
                                            L.L.C., its General Partner

                                            By: /s/ Chinh Chu
                                                --------------------------------
                                                Name: Chinh Chu
                                                Title: Member

                                    BLACKSTONE FAMILY INVESTMENT PARTNERSHIP
                                    IV-A L.P.

                                        By: Blackstone Management Associates IV
                                            L.L.C., its General Partner

                                            By: /s/ Chinh Chu
                                                --------------------------------
                                                Name: Chinh Chu
                                                Title: Member

                                        APOLLO INVESTMENT FUND V, L.P.

                                        By: Apollo Advisors V, L.P.
                                            Its general partner

                                        By: Apollo Capital Management V, Inc.
                                            Its general partner

                                        By: /s/ Joshua Harris
                                            ------------------------------------
                                            Name:  Josh Harris
                                            Title: Vice President

                                        APOLLO/NALCO ACQUISITION LLC

                                        By: Apollo Management V,L.P., its
                                            Manager

                                        By: AIF V Management, Inc.,
                                            its General Partner

                                        By: /s/  Joshua Harris
                                            ------------------------------------
                                            Name:  Josh Harris
                                            Title: Vice President

                                        AP NALCO LP

                                        By: Apollo Advisors V, L.P.,
                                            its General Partner

                                        By: Apollo Capital Management V, Inc.,
                                            its General Partner

                                        By: /s/  Joshua Harris
                                            ------------------------------------
                                            Name:  Josh Harris
                                            Title: Vice President

                                    GOLDMAN SACHS DIRECT INVESTMENT FUND
                                    2000, L.P.

                                        By:  GS Employee Funds 2000 GP, L.L.C.,
                                             its General Partner

                                             By: /s/ Sanjeev K. Mehra
                                                 -------------------------------
                                                 Name:
                                                 Title:

                                    GS CAPITAL PARTNERS 2000, L.P.

                                        By:  GS Advisors 2000, L.L.C.,
                                             its General Partner

                                             By: /s/ Sanjeev K. Mehra
                                                 -------------------------------
                                                 Name:  Sanjeev K. Mehra
                                                 Title:

                                    GS CAPITAL PARTNERS 2000 OFFSHORE, L.P.

                                        By:  GS Advisors 2000, L.L.C.,
                                             its General Partner

                                             By: /s/ Sanjeev K. Mehra
                                                 -------------------------------
                                                 Name:  Sanjeev K. Mehra
                                                 Title:

                                    GS CAPITAL PARTNERS 2000 GMBH & CO.
                                    BETEILIGUNGS KG

                                        By:  Goldman Sachs Management GP GmbH,
                                             its General Partner

                                             By: /s/ Sanjeev K. Mehra
                                                 -------------------------------
                                                 Name:  Sanjeev K. Mehra
                                                 Title:

                                    GS CAPITAL PARTNERS 2000 EMPLOYEE FUND, L.P.

                                        By:  GS Employee Funds 2000 GP, L.L.C.,
                                             its General Partner

                                             By: /s/ Sanjeev K. Mehra
                                                 -------------------------------
                                                 Name:  Sanjeev K. Mehra
                                                 Title:

                                    NH ACQUISITION LLC

                                        By: GS Advisors 2000, L.L.C.,
                                            its Manager

                                            By: /s/ Sanjeev K. Mehra
                                                --------------------------------
                                                Name:  Sanjeev K. Mehra
                                                Title:
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