PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. Ill 



Ki = rii + O. {t, 71^, V2, . . . ??3 n» ^IJ ^2J • • • ^3n). 1 



introducing 6 n varying elements «,. \., of which the set X,. would have been represented 

 in our recent notation as follows : 



^i = «3n + »; (73.) 



5j c> tv * jv - 



we see that all the partial differential coefficients of the forms —J, —J, *, f, vanish 



5tj^ SsB-y drj^ Sxsr^ 



when ^ = 0, except the following* : 



T^=''8^=" (74.) 



and, therefore, that when t is made = 0, in the coefficients a^^^, (59.), all those coeffi- 

 cients vanish, except the following : 



^r, 3n f r = 1^ ^3n + r,r = ~~ 1 (75.) 



But it has been proved that these coefficients «,^ ,, when expressed as functions of 

 the elements, do not contain the time explicitly ; and the supposition t = introduces 

 no relation between those 6 n elements x- X^, which still remain independent : the co- 

 efficients «^„ therefore, could not acquire the values 1,0, —1, by the supposition 

 / = 0, unless they had those values constantly, and independently of that supposition. 

 The differential equations of the forms (B^), may therefore be expressed, for the pre- 

 sent system of varying elements, in the following simpler way ; 



dt ^ ^\ ' U'^ 8«i ' ^^ '^ 



and an easy verification of these expressions is offered by the formula (E^), which 

 takes now this form, 



^(^'^ + ^^)=0 .(H., 



17. The initial values of the varying elements «^ X^ are evidently e^pi, by the defi- 

 nitions (72.), and by the identical equations (71) ; the problem of integrating rigo- 

 rously the equations of disturbed motion (G.), between the variables rji ts^ and the 

 time, or of determining these variables as functions of the time and of their own 

 initial values e,- /?,., is therefore rigorously transformed into the problem of integrating 

 the equations (G^), or of determining the 6 n elements »^ X, as functions of the time 

 and of the same initial values. The chief advantage of this transformation is, that if 

 the perturbations be small, the new variables (namely, the elements,) alter but little : 

 and that, since the new differential equations are of the same form as the old, they 

 may be integrated by a similar method. Considering, therefore, the definite integral 



