TRANSACTIONS OF THE SKCTIONS. 515 



but we know its variation (2.), taken with respect to the 6 n co- 

 ordinates ; and on account of the independence of their 6 n va- 

 riations, we can resolve this expression (2.) into two groups, 

 containing each 3 n equations : namely, 



S I 8 S I S S 1 /(. s 



— = im x'i , -^ = mi y'i, ^ = mi z'l . . [p.) 

 oxi cyi Zi 



and 



—- = — 7ni a'i , ^-j = —tHi b'i , -_ = — tm c'i ; . (6.) 

 Ui 6 Oi d 



the first members being partial differential coefficients of the 

 function S, which Mr. Hamilton calls the Principal Function 

 of motion of the attracting or repelling system. He thinks 

 that if analysts had perceived this principal function S, and 

 these groups of equations (5.) and (6.), they must have per- 

 ceived their importance. For the group (5.) expresses the 3 n 

 intermediate integrals of the known equations of motion (4.), 

 under the form of 3 /« relations between the time t, the masses 

 m, the varying coordinates x, y, z, the varying components of 

 velocities x' y' s', and the 3 n initial constants a b c; while the 

 group (6.) expresses the 3 n final integrals of the same known 

 differential equations, as 3 ra relations, with 6n initial and ar- 

 bitrary constants a b c a' b' c', between the time, the masses, 

 and the 3 n varying coordinates. These 3 n intermediate and 

 3 n final integrals, it was the problem of dynamics to discover. 

 Mathematicians had found seven intermediate, and none of the 

 final integrals. 



Professor Hamilton's solution of this long celebrated pro- 

 blem contains, indeed, one unknown function, namely the 

 princijjal function S, to the search and study of which he 

 has reduced mathematical dynamics. This function must not 

 be confounded with that so beautifully conceived by Lagrange, 

 for the more simple and elegant expression of the known dif- 

 ferential equations. Lagrange's function s^a^e*, Mr. Hamilton's 

 function would solve the problem. The one serves to form the 

 differential equations of motion, the other would give their 

 integrals. To assist in pursuing this new track, and in dis- 

 covering the form of this new function, Mr. Hamilton remarks 

 that it must satisfy the following partial differential equation 

 of the first order and second degree, (the time being now made 

 to vary,) 



Lm'-m'-oi=-'-^-'-^ 



^ t >im \_ \o x/ \^ y 



2 l2 



