PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



303 



".• = P2 (^iJ l^i> Tiy ^'i, P'iy y'i^ t)^ 



Ki — h («»» Piy yi> ^'i> l^'i^ y'i» ; 



we shall know that the following relations are rigorously and identically/ true, 



Sw^') 8w;('^ 8to^'^ iw^'^' 



(A^) 



^i = ^1 y^ii Pi, 7iy - T^, 



8/3.' 





"J^i S^ 



_ /" /2 ?^ ?^ i!^^ 8to^'\ 



h — ?'2 V<*i^ ft, Yi, — g «. J — S/3. ' ~ 8 y. ' §„(«)/' 



^i = <P3 y^iy Pi, Yi, - T^y 



diso^^ 8w^'^ 8^^') 8w)('^ 



J) 8to('K 

 ?8/^V' J 



(B^) 



and consequently that these relations will still be rigorously true when we substitute 

 for the four coefficients of w^^ their rigorous values (X^'.) and (Z<'.) for the case of a 

 multiple system. We may thus retain in rigour for any multiple system the final in- 

 tegrals (A^.) of the motion of a binary system, if only we add to the initial com- 

 ponents oc'i ^'i y'i of relative velocity, and to the time t, the following perturbational 

 terms : 



m, 



dw^'^ 



A a!. —2^^ g^ 



m. 



ISO 



k 



1 ^^n . ±2 ^ 

 m '8a.' 



n I 



8V,. 



' 7». 8 «. ' 



A /?' — ^ * LTL- _4_ -1 ^^'^ _L J- 5 



^Pi- ^" • m, + m„ 8/3, ^ m, 86. ^ m ^' 



m. 



'ISO 



(k) 



1 8V 





8/3.' 



and 



^ y «■ — ^/' • w^ + w„ Sy,. "^ /». 87. "^ m^ ^' 8y 



A*=-^ 



(C-.) 



8H,' 



(D'-) 



In the same way, if the theory of binary systems, or the elimination of g^'^ between 

 the four equations (F.) (L^.)? has given three intermediate integrals, of the form 



< = •4'2(^i^^i»?i^«i»f3.,y.,0, V ....... (Eg 



we can conclude that the following equations are rigorous and identical, 



1: = -^1 (ip ^i» Q. «i' f3., y., ^ j , 



I o 



r = -^2 (^,> ip <i, ojp Pi, 7i, fpj, > 



y^ = '^'a (^t,, ^,, ^p «p f3., 7., ^^y , 



8 

 8„, 



(F-.) 



2 R 2 



