PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS, 

 ei = pji — TSTi t, 62 = ^12 — ^2 t> ^3 



125 



Pi 



m 



(148.) 



V 



P2 = ^^2^ /?3 = ^3 + ^ ^- J 



We shall then have rigorously, for the six disturbed variables jj^ V2 % ^i ^2 ^-a ex- 

 pressions of the same forms as in the integrals (127.) and (128.) of undisturbed 

 motion, but with variable instead of constant elements, namely, the following : 



;?! = »1 + Xi t, f}2 = ^2 + h*> ^3 = »3 + h^ — -Q g t^ 



ro-j = Xi, 



TSTn 



^2> 



= h- g^'* 



(149.) 



and the rigorous determination of the six varying elements x^ »2 «3 ^1 ^2 ^3? as func- 

 tions of the time and of their own initial values e^ e^ e^ p^ p^ p^, depends on the in- 

 tegration of the 6 following equations, in ordinary differentials of the first order, of 

 the forms (105.): 



^ = TIT = ^^ (*2 + 5^2 t), 



and 



(150.) 



= — (0,2 («i -f- Xi /), 

 = — ^2 (;p^ _|, ^2 ^), 





^l 



(151.) 



H2 being here the expression 



H2 =^{K + ?^lO'+ (^2 + ^2 0^} + y(^3 + hf-^g^'Y^ 



(152.) 



which is obtained from (125.) by substituting for the disturbed coordinates ri^ ri2 fj^ 

 their values (149.), as functions of the varying elements and of the time. It is not 

 difficult to integrate rigorously this system of equations (150.) and (151.) ; and we 

 shall soon have occasion to state their complete and accurate integrals : but we shall 

 continue for a while to treat these rigorous integrals as unknown, that we may take 

 this opportunity to exemplify our general method of indefinite approximation, for all 

 such dynamical questions, founded on the properties of the functions of elements C 

 and E. Of these two functions either may be employed, and we shall use here the 

 function C. 



29. This function, by (109.) and (152.), may rigorously be expressed as follows : 



