434 Mr. J. H. Cotterill on the Further Application 



Now if with M. Lame* we take 



we have 



N a + N 8 =(3X + 2/*)0-N 1 , 

 and the first equation becomes 



\ y</ fx /jl dec 



or 



The two other equations may be reduced in like manner; also 

 from the second set of equations, 



T _/* f#2 ,#3"! 



ll ~2\~dF + TfJ } 



with two other symmetrical equations. 



Now on comparing these equations with M. Lame's (p. 65, 

 equations (1)), they are seen to be identical on supposing <£ 1 = 2w, 

 (j) 2 =z2v, <f> 3 =2w; and thus it appears that the results of the ap- 

 plication of the principle of Least Action are identical with those 

 obtained by M. Lame, which furnishes an independent proof of 

 the principle in the case of perfect elasticity. 



Before leaving the subject, I will make some general remarks 

 in conclusion. 



Though nothing has been strictly proved, except that the va- 

 riation in the work done in a perfectly elastic body due to a 

 change in the resisting forces is zero, yet it is sufficiently evident 

 that the work is a minimum. That is, of all values of the work 

 possible on the supposition that the resistances in terms of which 

 the work is expressed may be not merely resisting or (as they 

 have been called) "passive" forces, but applied or " active" 

 forces, the least is that which corresponds to the passive forces. 

 I think that, seen from this point of view, the principle of Least 

 Action is nearly self-evident, even without the confirmatory evi- 

 dence that the variation in the case of a perfectly elastic body is 

 zero ; for the change of a force which is really an effect, into a 

 force which is really a cause, must increase directly or indirectly 

 the total effect. 



On the other hand, the energy expended may be said to be a 

 maximum. That is, of all values of the energy expended by the 

 really active forces possible on the supposition that those forces 

 may be active which are really passive, the greatest value physi- 

 cally possible corresponds to the passive forces. Thus a weight 



