<PAGE>

                                                                  Exhibit 4(B)-2

      Form of Resolutions of Duke Energy Gas Transmission Services, LLC,
                    the General Partner of the Registrant,
                  Establishing the Terms of a Series of Notes

          RESOLVED, that a series of unsecured senior notes of Texas Eastern
Transmission, LP under the Indenture dated as of December 1, 2000, as
supplemented (the "Indenture") between Texas Eastern Transmission, LP and The
Chase Manhattan Bank, as Trustee (the "Trustee"), shall be established and shall
be issued and designated as Texas Eastern Transmission, LP's   % Notes due
(the"Notes");

          FURTHER RESOLVED, that in addition to the terms provided in the
Indenture, the terms of the Notes shall be as follows:

          (1) The aggregate principal amount of the Notes shall be $     , and
          no further Notes shall be authenticated and delivered except as
          provided by Section 304, 305, 306, 906 or 1106 of the Indenture;

          (2) The Stated Maturity of the principal of the Notes shall be
              ,       ;

          (3) The Notes shall bear interest at the rate of  % per annum from
                    ,        , or from the most recent Interest Payment Date
          to which interest has been paid or duly provided for;

          (4) The Interest Payment Dates with respect to the Notes shall be
          and     in each year, commencing         ,   , and the Regular Record
          Date for interest payable on any such Interest Payment Date shall be
          the fifteenth calendar day (whether or not a Business Day) prior to
          the relevant Interest Payment Date;

          (5) Payments of principal and interest on Notes represented by a
          Global Security or Global Securities shall be made by wire transfer of
          immediately available funds to the Holder of such Global Security or
          Global Securities; provided, however, that if any of the Notes are no
          longer represented by a Global Security,

                                       1
<PAGE>

          payments of principal and interest on such Notes shall be made at the
          Corporate Trust Office of the Trustee or at such other office or
          agency maintained by Texas Eastern Transmission, LP for such purpose;
          provided that payments of interest may be made, at the option of Texas
          Eastern Transmission, LP, by check mailed to the address of the Person
          entitled thereto as such address shall appear in the Security Register
          or by wire transfer at such place and to such account at a banking
          institution in the United States as may be designated in writing to
          the Trustee at least sixteen (16) days prior to the date for payment
          by the Person entitled thereto. The Notes shall be exchangeable, and
          transfers thereof shall be registrable, at the office of the Trustee
          located at 55 Water Street, New York, NY 10041 or at any other office
          or agency of Texas Eastern Transmission, LP maintained for such
          purpose;

          (6) The Notes shall not be subject to redemption and shall not be
          subject to a sinking fund requirement; and

          (7) The Notes shall be issued in the form of one or more Global
          Securities, with The Depository Trust Company as initial Depositary;

          FURTHER RESOLVED, that the Notes shall be issued in fully registered
form without coupons in denominations of $1,000 and integral multiples thereof;

          FURTHER RESOLVED, that the Trustee is authorized to act on behalf of
Texas Eastern Transmission, LP as initial Paying Agent for the Notes;

          FURTHER RESOLVED, that the form of the Notes set forth in Exhibit A
hereto be, and hereby is, approved;

          FURTHER RESOLVED, that           and               are selected as
the representatives of the several underwriters (the "Underwriters") to which
the Notes shall be sold; that the purchase price to be paid to Texas Eastern
Transmission, LP by the Underwriters shall be   % of the principal amount of the
Notes; and that the initial public offering price shall be   % of the principal
amount of the Notes; and

          FURTHER RESOLVED, that the underwriting agreement (the "Underwriting
Agreement") between Texas Eastern Transmission, LP and the Underwriters dated
          ,      , which is substantially in the form filed with the
Registration Statement on Form S-3 of Texas Eastern Transmission, LP filed with
the Securities and Exchange Commission on May 17, 2001, be and hereby is
approved, and the proper officers of Duke Energy Gas Transmission Services, LLC,
the general partner of Texas Eastern Transmission, LP, be and hereby are
authorized and empowered to execute such Underwriting Agreement on behalf of
Texas Eastern Transmission, LP.

<PAGE>

                                                                       EXHIBIT A

                                    FORM OF
                                    % NOTE DUE

No.                                                                    CUSIP No.

                        TEXAS EASTERN TRANSMISSION, LP
                                   % NOTE DUE

Principal Amount:

Regular Record Date: close of business on the 15th calendar day prior to the
                     relevant Interest Payment Date

Original Issue Date:             ,

Stated Maturity:                 ,

Interest Payment Dates:             and           , commencing

Interest Rate:              % per annum

Authorized Denomination: $    or any integral multiple thereof

Texas Eastern Transmission, LP, a Delaware limited partnership (the "Company,"
which term includes any successor under the Indenture referred to on the
reverse hereof), for value received, hereby promises to pay to, or registered
assigns, the principal sum of              DOLLARS ($           ) on the Stated
Maturity shown above and to pay interest thereon from the Original Issue Date
shown above, or from the most recent Interest Payment Date to which interest has
been paid or duly provided for, semiannually in arrears on each Interest Payment
Date as specified above, commencing on the Interest Payment Date next succeeding
the Original Issue Date shown above and on the Stated Maturity at the rate per
annum shown above (the "Interest Rate") until the principal hereof is paid or
made available for payment and on any overdue principal and on any overdue
installment of interest. The interest so payable, and punctually paid or duly
provided for, on any Interest Payment Date will, as provided in the Indenture,
be paid to the Person in whose name this   % Note due      (this "Security") is
registered on the Regular Record Date as specified above next preceding such
Interest Payment Date. Except as otherwise provided in the Indenture, any such
interest not so punctually paid or duly provided for will forthwith cease to be
payable to the Holder on such Regular Record Date and may either be paid to the
Person in whose name this Security is registered at the close of business on a
Special Record Date for the payment of such Defaulted Interest to be fixed by
the Trustee, notice whereof shall be given to Holders of Securities of this
series not less than 10 days prior to such Special Record Date, or be paid at
any time in any other lawful manner not inconsistent with the requirements of
any securities exchange, if any, on which the Securities of
<PAGE>

this series shall be listed, and upon such notice as may be required by any such
exchange, all as more fully provided in the Indenture.

Payments of interest on this Security will include interest accrued to but
excluding the respective Interest Payment Dates. Interest payments on this
Security shall be computed and paid on the basis of a 360-day year of twelve 30-
day months. In the event that any date on which interest is payable on this
Security is not a Business Day, then payment of the interest payable on such
date will be made on the next succeeding day that is a Business Day (and without
any interest or payment in respect of any such delay) with the same force and
effect as if made on the date the payment was originally payable. "Business Day"
means a day other than (i) a Saturday or a Sunday, (ii) a day on which banking
institutions in New York, New York are authorized or obligated by law or
executive order to remain closed or (iii) a day on which the Corporate Trust
Office is closed for business.

Payments of principal of, premium, if any, and interest on the Securities of
this series shall be made in such coin or currency of the United States of
America as at the time of payment is legal tender for payment of public and
private debts. Payments of principal of, premium, if any, and interest on
Securities of this series represented by a Global Security shall be made by wire
transfer of immediately available funds to the Holder of such Global Security;
provided that, in the case of the payment of principal and premium, if any, such
Global Security is first surrendered to the Paying Agent. If any of the
Securities of this series are no longer represented by a Global Security,
payments of principal, premium, if any, and interest on such Securities shall be
made at the office of the Paying Agent maintained for such purpose; provided
that payments of interest may be made, at the option of the Company, (A) by
check mailed to the address of the Person entitled thereto as such address shall
appear in the Security Register or (B) by wire transfer at such place and to
such account at a banking institution in the United States as may be designated
in writing to the Trustee at least sixteen (16) days prior to the date for
payment by the Person entitled thereto.

REFERENCE IS HEREBY MADE TO THE FURTHER PROVISIONS OF THIS SECURITY SET FORTH ON
THE REVERSE HEREOF, WHICH FURTHER PROVISIONS SHALL FOR ALL PURPOSES HAVE THE
SAME EFFECT AS IF SET FORTH AT THIS PLACE.

Unless the certificate of authentication hereon has been executed by the Trustee
by manual signature, this Security shall not be entitled to any benefit under
the Indenture or be valid or obligatory for any purpose.
<PAGE>

IN WITNESS WHEREOF, the Company has caused this instrument to be duly executed.

Dated:

                               TEXAS EASTERN TRANSMISSION, LP



                               By: Duke Energy Gas Transmission Services, LLC,
                                    as General Partner

                               By:
                                   -------------------------------------

Attest:

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

                          CERTIFICATE OF AUTHENTICATION

          This is one of the Securities of the series designated therein
referred to in the within-mentioned Indenture.


                               THE CHASE MANHATTAN BANK,
                                as Trustee


                               By:
                                   -------------------------------------
<PAGE>

                          (Reverse Side of Security)

This Security is one of a duly authorized issue of Securities of the Company
(the "Securities"), issued and issuable in one or more series under an
Indenture, dated as of December 1, 2000 (the "Indenture"), between the Company
and The Chase Manhattan Bank, as Trustee (the "Trustee," which term includes any
successor trustee under the Indenture), to which Indenture and all indentures
supplemental thereto reference is hereby made for a statement of the respective
rights, limitation of rights, duties and immunities thereunder of the Company,
the Trustee and the Holders of the Securities issued thereunder and of the terms
upon which said Securities are, and are to be, authenticated and delivered. This
Security is one of the series designated on the face hereof as    % Notes due
       in the aggregate principal amount of up to $           . Capitalized
terms used herein for which no definition is provided herein shall have the
meanings set forth in the Indenture.

The Securities of this series are not subject to redemption and are not subject
to a sinking fund requirement.

If an Event of Default with respect to the Securities of this series shall occur
and be continuing, the principal of the Securities of this series may be
declared due and payable in the manner, with the effect and subject to the
conditions provided in the Indenture.

The Indenture permits, with certain exceptions as therein provided, the
amendment thereof and the modification of the rights and obligations of the
Company and the rights of the Holders of the Securities of all series affected
under the Indenture at any time by the Company and the Trustee with the consent
of the Holders of not less than a majority in principal amount of the
Outstanding Securities of all series affected thereby (voting as one class). The
Indenture contains provisions permitting the Holders of not less than a majority
in principal amount of the Outstanding Securities of all series with respect to
which a default under the Indenture shall have occurred and be continuing
(voting as one class), on behalf of the Holders of the Securities of all such
series, to waive, with certain exceptions, such default under the Indenture and
its consequences. The Indenture also permits the Holders of not less than a
majority in principal amount of the Securities of each series at the time
Outstanding, on behalf of the Holders of all Securities of such series, to waive
compliance by the Company with certain provisions of the Indenture affecting
such series. Any such consent or waiver by the Holder of this Security shall be
conclusive and binding upon such Holder and upon all future Holders of this
Security and of any Security issued upon the registration of transfer hereof or
in exchange hereof or in lieu hereof, whether or not notation of such consent or
waiver is made upon this Security.

No reference herein to the Indenture and no provision of this Security or of the
Indenture shall alter or impair the obligation of the Company, which is absolute
and unconditional, to pay the principal of and interest on this Security at the
times, place and rate, and in the coin or currency, herein prescribed.

As provided in the Indenture and subject to certain limitations therein set
forth, the transfer of this Security is registrable in the Security Register,
upon surrender of this Security for registration of transfer at the office or
agency of the Company for such purpose, duly endorsed by, or accompanied by a
written instrument of transfer in form satisfactory to the Company and the
<PAGE>

Security Registrar and duly executed by, the Holder hereof or his attorney duly
authorized in writing, and thereupon one or more new Securities of this series,
of authorized denominations and of like tenor and for the same aggregate
principal amount, will be issued to the designated transferee or transferees. No
service charge shall be made for any such registration of transfer or exchange,
but the Company may require payment of a sum sufficient to cover any tax or
other governmental charge payable in connection therewith.

As provided in and subject to the provisions of the Indenture, the Holder of
this Security shall not have the right to institute any proceeding with respect
to the Indenture or for the appointment of a receiver or trustee or for any
other remedy thereunder, unless such Holder shall have previously given the
Trustee written notice of a continuing Event of Default with respect to the
Securities of this series, the Holders of not less than a majority in principal
amount of the Securities of this series at the time Outstanding shall have made
written request to the Trustee to institute proceedings in respect of such Event
of Default as Trustee and offered the Trustee reasonable indemnity, and the
Trustee shall not have received from the Holders of a majority in principal
amount of Securities of this series at the time Outstanding a direction
inconsistent with such request and shall have failed to institute any such
proceeding for 60 days after receipt of such notice, request and offer of
indemnity. The foregoing shall not apply to any suit instituted by the Holder of
this Security for the enforcement of any payment of principal hereof or any
premium or interest hereon on or after the respective due dates expressed
herein.

The Indenture contains provisions for defeasance at any time of the entire
indebtedness of the Securities of this series and for covenant defeasance at any
time of certain covenants in the Indenture upon compliance with certain
conditions set forth in the Indenture.

Prior to due presentment of this Security for registration of transfer, the
Company, the Trustee and any agent of the Company or the Trustee may treat the
Person in whose name this Security is registered as the owner hereof for all
purposes, whether or not this Security be overdue, and neither the Company, the
Trustee nor any such agent shall be affected by notice to the contrary.

The Securities of this series are issuable only in registered form without
coupons in denominations of $         and any integral multiple thereof. As
provided in the Indenture and subject to the limitations therein set forth,
Securities of this series are exchangeable for a like aggregate principal amount
of Securities of this series of a different authorized denomination, as
requested by the Holder surrendering the same upon surrender of the Security or
Securities to be exchanged at the office or agency of the Company.

This Security shall be governed by, and construed in accordance with, the
internal laws of the State of New York.
<PAGE>

                                 ABBREVIATIONS

The following abbreviations, when used in the inscription on the face of this
instrument, shall be construed as though they were written out in full according
to applicable laws or regulations:

TEN COM -- as tenants in common     UNIF GIFT MIN ACT --       Custodian
                                                        -------         --------
                                                        (Cust)          (Minor)


TEN ENT -- as tenants by the entireties           under Uniform Gifts to Minors
                                                  Act
                                                     ---------------------------
                                                                (State)

JT TEN -- as joint tenants with rights of
          survivorship and not as tenants
          in common

Additional abbreviations may also be used though not on the above list.



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FOR VALUE RECEIVED, the undersigned hereby sell(s) and transfer(s) unto_______
(please insert Social Security or other identifying number of assignee)

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PLEASE PRINT OR TYPEWRITE NAME AND ADDRESS, INCLUDING POSTAL ZIP CODE OF
ASSIGNEE

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the within Security and all rights thereunder, hereby irrevocably constituting
and appointing


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agent to transfer said Security on the books of the Company, with full power of
substitution in the premises.


Dated:
      ------------------------       -------------------------------------------


                                     -------------------------------------------
                                     NOTICE: The signature to this assignment
                                     must correspond with the name as written
                                     upon the face of the within instrument in
                                     every particular without alteration or
                                     enlargement, or any change whatever.