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These securities were offered par-for-par to holders of the debt issued by the Congress during the Revolutionary War and the Confederation and to holders of state debts incurred for national purposes Holders of Congressional debt received two thirds of principal in the 6s of 1790 and one third in the Deferred 6s of 1790, and thus received average interest of 4% to 1800 and 6% thereafter Their claims for unpaid interest were met with the 3s of 1790 par-for-par Holders of approved state debts received four ninths in the 6s of 1790, two ninths in the Deferred 6s of 1790, and three ninths in the 3s of 1790 The securities accepted in exchange for these new issues had sold at large discounts Therefore, a market rate of interest cannot be inferred for the years 1790 1794 from these nominal rates The conversion was not forced.

Carothers derived a relationship between the extent of reaction at the gel point and the average functionality favg of the polymerization system for the case where the two functional groups A and B are present in equivalent amounts [Carothers, 1936]. The derivation follows in a manner similar to that for Eq. 2-78. The average functionality of a mixture of monomers is the average number of functional groups per monomer molecule for all types of monomer molecules. It is de ned by

favg Ni fi Ni 2-134

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where Ni is the number of molecules of monomer i with functionality fi, and the summations are over all the monomers present in the system. Thus for a system consisting of 2 mol of glycerol (a triol) and 3 mol of phthalic acid (a diacid) there is a total of 12 functional groups per ve monomer molecules, and favg is 12 or 2.4. For a system consisting of a equimolar 5 amounts of glycerol, phthalic anhydride, and a monobasic acid, there is a total of six functional groups per three molecules and favg is 6 or 2. 3 In a system containing equivalent numbers of A and B groups, the number of monomer molecules present initially is N0 and the corresponding total number of functional groups is N0 favg . If N is the number of molecules after reaction has occurred, then 2 N0 N is the number of functional groups that have reacted. The extent of reaction p is the fraction of functional groups lost:

It was generally believed that the market rate of interest would fall as the public credit rose (493) All of the old debt was redeemable at the pleasure of the government, whereas the new 6s could not be redeemed faster than 2% a year; therefore the conversion appealed to the self-interest of creditors The national revenues and proceeds of the sale of Western lands were pledged to the payment of interest on these domestic issues subject only to the servicing of the $12 million of foreign debt, which was always regarded as having a prior claim During the period 1791 1801 Treasury surpluses exceeded deficits In 1792 a regular sinking fund was established by Hamilton as a device to improve the credit of the government and raise the price of its securities (494) Appropriations were made and a commission appointed to borrow money and make purchases in the market.

p 2 N0 N N0 favg 2-135

FIGURE C-21

Xn N0 N 2-136

It ought to be the policy of the government, said Hamilton, to raise the value of stock to its true standard, as fast as possible When it arrives at that point, foreign speculations will become beneficial While few market quotations on the new government securities are available until 1798, Hamilton s statement implies that the 6% bonds stood below par in their earlier years A price of 70 is reported for 1791, to yield 857% The favorable financial position of the Treasury was changed in 1797 1799 when an undeclared naval war with France required the creation of a military establishment A new loan was authorized in 1798 It was sold for cash in 1799 and 1800 and bore the highest.

Xn 2 2 p favg 2-137

p 2 2 favg X n favg 2-138

Equation 2-138, often referred to as the Carothers equation, relates the extent of reaction and degree of polymerization to the average functionality of the system. An important consequence of Eq. 2-138 is its limiting form at the gel point where the number-average degree of polymerization becomes in nite. The critical extent of reaction pc at the gel point is given by

nominal rate for a long-term government bond in our history until recently: 1799 $50 million 8s of 1798, redeemable in 1809;* sold at 100 = 8% 1800 $15 million 8s of 1800, redeemable in 1809;* sold at 1053 4 = 734% Market prices for government bond issues are reported from 1798 (495) In 1798 1799, as Table 39 reveals, the market for federal government bonds was depressed This was ascribed to public expectation of war and invasion The 6s of 1790 sold as low as 75, to yield 8% These prices explain the Treasury s use of a rate as high as 8% for its new issues in 1799 and 1800 In 1798 the 3s of 1790 were selling in a range of 421 2 50, to provide a current yield of 6 707%.

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