134 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



the six quantities zX (jij v r m being constant for the undisturbed motion of any one 

 binary system ; and therefore the six functions x^^\ X^^\ X^^\ X^'^\ %^^\ X^^\ or «, K, ^^ 

 V, r, u, being such as to satisfy identically the following equation, 



8x SxSHi Sj^SHi SxSHi l!i LMl I !^ ?ii liiilL n2\ 



O — mg^ + g^-g^-g^ g^ +g^ ^j^i - y 8, +8? 8z' ""8 2' 8? ^ ' ^^ ■> 



with five other equations analogous, for the five other elements K, yij, v, r, a, in any one 

 binary system (m, M). 



33. Returning now to the original multiple system, we may retain as definitions 

 the equations (K^.), but then we can no longer consider the elements «. X. uj. v. r. u. of 



the binary system (m., M) as constant, because this system is now disturbed by the 

 other masses w^ ; however, the Qn — Q equations of disturbed relative motion, when 

 put under the forms 



SH. 



^ 



8^ ' 



8^, 



8,, ^ 



8H. 



(M2.) 



8? 



and combined with the identical equations of the kind (L^.), give the following simple 

 expression for the differential of the element », in its disturbed and variable state, 



'^ dt~- l^ Sy 8^ 8^ + 8>j 8y 8y 8>, + 8?' 82' 82' 8? ^ * ^^ 'f 



together with analogous expressions for the differentials of the other elements. And 

 if we express | ;? ^ a?' t/' ;s', and therefore H2 itself, as depending on the time and on 

 these varjring elements, we may transform the ^n^^ differential equations of the 

 1st order, (M^.), between ^yiZ^x' y' z' t, into the same number of equations of the 

 same order between the varying elements and the time ; which will be of the forms 



^x f , , 8 H2 , _ 8 H^ , , . 8 H2 , ^ 8 H2 , ^ , 8 H2 1 



:^= K ?^} -gX + ^*^^>-87- + ^*' "> "87" + ^^^ '■>~87" + ^^^ "^T^' 



d^ ,, , 8 H, . ,. , 8 H2 , . 8 H2 , .^ . 8 H2 . ,. . 8 Ho 



m 



^^ r . 8 H2 , ^ , 8 H2 , ^ 8 Hg . ^ , 8 H. . , 8 Ho 



'^-df = i'> *> 17-+ (^ ^} -87- + ^"^ ^> -8^+ ^^ ^> -87- + ^^ ^> T^' 



r» ^'' / 1 ^ ^2 , . . , 8 H2 , ^ 8 H2 , ^ 8 H2 , , , 8 H2 



>(02.) 



Wl 



<i« _ , S H2 . . , 1 S H2 , , . 8 H2 , , , 8 H2 , ^ 8 Hi 



