PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 263 



= 'p2Qll,^2,"'%n + ky [ (31.) 



giving k of the new marks of position as functions of the remaining 3 n. 



^. 



3n+ 1 



'4^iQii,n2,'"^sny IN 



'^3n+2 = '^2('Jl. ^2. •••^3n> J^ (32.) 



the expression 



T = l2.m(^'2 + y2 + ;2'2), . (4.) 



will become, by the introduction of these new variables, a homogeneous function of 

 the second dimension of the 3n -\- k rates of increase fj\, rj^, . . . ri'^^.j^, involving also 



in general rj^^ 9)2^ . . . yia„_L.f,i and having a variation which may be thus expressed : 





(33.) 



or in this other way, 

 ST 



8T 



ST = ^S^\ + F3-^'j'2 + ..- + 



8T 



,'. -^ ■ 8V. "'^^•••^8V3„ "3n 



+ 



ST 





(34.) 



Ml " '1-2 - -13 n 



on account of the relations (32.), which give, when differentiated with respect to the 

 time, 





3n d fj 



3n 





'3n 



> 



(35.) 





'3n -' 



and therefore, attending only to the variations of quantities of the form >?', 



H, 



H, 



SrJ,, 



'3n + 2 Jji 'A ' 6>J2 ■* ' oy] '3n' 



1 



> 



3n + * 8,,j S)J2 ^ Sijg^ •*" J 



2 M 2 



(36.) 



