PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



305 



w — 2 auxiliary quantities of the form ^W ; and then we are to consider those initial 

 components and auxiliary quantities and the time, as depending themselves on the 

 initial and final coordinates, and on H^. But it is not difficult to prove, by the fore- 

 going principles, that when t and g^^) are thus considered, their variations are, in the 

 present order of approximation. 



t — 







(K^) 



and 





'82w<*)\-i 



the sign of variation I, referring only to the initial and final coordinates ; and also 

 that 



^'W^) Hi 



8^ to(0 S ^i 



Bgd)^ 



^t 



+ 



82^(i) 



Hi_, 



8«t4)(0 Hi 



(W.) 



along with two other analogous relations between the coefficients of the two other 

 coordinates n'\ C *^ ; f^'^m which it follows that t and g^ , and therefore a!^ j8'^ y' may 

 be treated as constant, in taking the variation of the disturbing part V^2j for the pur- 

 pose of calculating the perturbations (H^.) : and that the terms involving A t are 

 destroyed by other terms. We may therefore put simply 



At ^^i A ' I ^ ^» A /a' I ^ ^i A ' 



^c 



8/3', 



A,, = ^ A«, + ^Ap, + ^-^Ay„ 

 A 2- = ^A A a'. 4- ^ A 3'. + ^ii-A y'. 



> 



(N7.) 



employing for A a'^ the following new expression, 



^y'i /•'8R0'''^) 



A<=2,.m,{7^' 8«, 



'^ 8 R(*"> ^) 7 . . S a'i /»f 8 R(»' *) 



8 a' 



rf# 



8«j^o «yi J J 



dt 

 dt 



(Or.) 



8/3', ' 8«i^o Sy^ 



together with analogous expressions for A |3'-, A /•, in which the sign of summation 2^, 

 refers to the disturbing masses, and in which the quantity 



+ 



gW*) grW gz-W 



(F.) 



'^ s^* ^'« H ^^' n, 



is considered as depending on a, jS^ y^ a', |3^ /> a^^. (3^ y^ a'^ |3'^ y'^ t, by the theory of bi- 

 nary systems, while a'. |8'. y'. are considered as depending, by the same rules, on 

 a. 3- 7- 1 ^ C- and #. 



