PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 297 



all the w — 1 first masses vanish, with the exception of each successively ; namely, to 

 the part 



2T/" = -^(ri2 + >/,^ + ^,2), (132.) 



when only mj, m„, do not vanish ; the part 



2T/^' = ^^(?? + '/2^ + CV), (133.) 



when all but Wg, w„, vanish ; and so on, as far as the part 



2 T/"-" = "'-■"" (g\_, + rl\_, + a-.)> • • • • (134.) 



which remains, when only the two last masses are retained. The sum of these w — 1 

 parts is not, in general, equal to the whole relative living force 2 T, of the system, 

 with all the n masses retained ; but it differs little from that whole when the first 

 n — \ masses are small in comparison with the last mass m„ ; for the rigorous value 

 of this difference is, by (68.), and by (132.) (133.) (134.), 



2 T, — 2 T/^^ _ 2 T ^^^ — . . . — 2 t/"""^^ = 





^. (136.) 



an expression which is small of the second order when the w — 1 first masses are 



small of the first order. If, then, we denote by V,, V/^ , . . . V/"~^ , the relative 

 actions, or accumulated relative living forces, such as they would be in the w — 1 

 binary systems, (m^ m^), (mg mj, . . . (m„_i m„), without the perturbations of the 

 other small masses of the entire multiple system of n points ; so that 



V/" =^2 T/" <f ;, V« =/'2 T/^' dt,... V/"-" =/'2 T/"-" dt, (Q5.) 



the perturbations being neglected in calculating these w — 1 definite integrals ; we 

 shall have, as an approximate value for the whole relative action V^ of the system, the 

 sum V^i of its values for these separate binary systems, 



V,i = V/'^ + V/'^ + . . . + V/"-'^ (R5; 



This sum, by our theory of binary systems, may be otherwise expressed as follows : 



(1) (2) (n) 



'^ m^ + m^ "" m^ + m^ »»n-i + »*» 



if we put for abridgement 



