PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 99 



and therefore 



$S' = 2(^Sg^---g^^;;), (21.) 



that is, by the equations of motion (A.), 



SS' = 2(^4^ + ^S,) = ^2.^S,; (22.) 



the variation of the integral S is therefore 



hS=^{7^hn —p^e), (23.) 



(p and e being still initial values,) and it decomposes itself into the following 6 n ex- 

 pressions, when S is considered as a function of the 6 ?i quantities fj. e., (involving also 

 the time,) 



5S as 



^1 - 8^ ; Pi = - fT,' 



__ 8S __ SS 



^^2 — 8,,^; ^2 — — 8^2 ' 



8S SS 



(B.) 



= 8« » Psn^— 8^^ ' 

 '3n 3ra 



which are evidently forms for the sought integrals of the 6 n differential equations of 

 motion (A.), containing only one unknown function S. The difficulty of mathema- 

 tical dynamics is therefore reduced to the search and study of this one function S, 

 which may for that reason be called the Principal Function of motion of a system. 

 This function S was introduced in the first Essay under the form 



S=X'{T + V)dt, 



the symbols T and U having in this form their recent meanings ; and it is worth 

 observing, that when S is expressed by this definite integral, the conditions for its 

 variation vanishing (if the final and initial coordinates and the time be given) are 

 precisely the differential equations of motion (3.), under the forms assigned by La- 

 grange. The variation of this definite integral S has therefore the double property, 

 of giving the differential equations of motion for any transformed coordinates when 

 the extreme positions are regarded as fixed, and of giving the integrals of those dif- 

 ferential equations when the extreme positions are treated as varying. 



5. Although the function S seems to deserve the name here given it of Principal 

 Function, as serving to express, in what appears the simplest way, the integrals of the 

 equations of motion, and the differential equations themselves ; yet the same analy- 

 sis conducts to other functions, which also may be used to express the integrals of 

 the same equations. Thus, if we put 



Q=/'(-2-W + H)'''' ^''-^ 



and take the variation of this integral Q without varying t ov dt, we find, by a simi- 

 lar process, , . 



lQ = ^(ti^r!r'-elp); (25.) 



o2 



