148 



PROFESSOR TAIT ON THE APPLICATION OF HAMILTON S 



still farther, by supposing the initial velocity to depend, according to some given 

 law, upon the coordinates of the initial point, and so forth. But such compli- 

 cations introduce analytical difficulties of the quasi-arithmetical kind merely, 

 not of a physical nature ; and we leave them to those who are curious in such 

 matters. 



4. In symbols, if t be the time of passing from x ,y Q ,z to z,y, z, we must 

 have 



ds 

 v 



a minimum: subject to the sole condition 



w 2 = 2(H-V) 



where H is the whole energy, and V the potential of the system of forces on 

 unit mass at the point x, y, z. 

 Hence, taking the variation, 



<x _ C(d8s dsdv\ 

 ~ J V v v' 2 )' 



But dsdSs = dxddx + dyddy + dzddz ; 



and vdv = 8(K - V) = X8x + Y8y + Z8z + §E, 



if X, Y, Z be the component forces on unit mass at x, y, z. Thus we have 



where the whole, integrated or not, is to be taken between the given limits. 



If the limits and the initial velocity be fixed, the first part of the expression 

 for St disappears ; and, that the integral may vanish, we must have 



*(~i) +" ss °' 



(A). 



with similar equations in y and z. This is simply the ordinary result given in 

 treatises on kinetics. 



But if we consider the effect of the alteration of the limits, or of the initial 

 energy, we have 



