of the Mechanical Theory of Heat. 171 



The expression containing the quantities h and i, which repre- 

 sents the work SL, is not a complete variation of a function of h 

 and i; on the other hand, if the equation be brought into the 

 following form, 



8L = a(^+281ogf) 



= h{$hgh + 28logi), 

 the two variations in the brackets can be reduced to one, viz. 



$L = hS(\ogh + 2\ogi), 

 or, otherwise written, 



SL = /*Slog(/*i 2 ) (23) 



Hence the work can be represented by the product of h and the 

 variation of a function of h and i, 



This result corresponds perfectly with equation (2) relative to 

 the theory of heat, 



dh=ChdZ. 



The quantity log (hi 2 ) is replaced in this equation by the pro- 

 duct CZ, in which C is a constant, and Z the magnitude which 

 in the theory of heat I have named the disgregation. We have 

 hence, so far as we wish to apply this conception to the station- 

 ary motion of a single point, arrived at a nearer determination 

 of it — namely, that the disgregation is proportional to the quan- 

 titylog(^ 2 ). 



8. In order to get an idea of the geometrical meaning of the 



quantity log (hi?), I will for h reintroduce the product - v%. We 



then obtain 



log (A?) = log (|t;«.i«) 



= log (v 2 .; 2 )-f logy 



= 2 log («V^) -flog™-. 



The last term on the right-hand side is invariable, and hence is 

 unimportant to equation (23), in which only the variation of the 

 quantity considered occurs ; we need therefore only attend to the 

 first term. 



Assuming now as a special case that the velocity is constant 

 (which occurs, for example, when a point moves in a circular 

 path round a fixed centre of attraction, or when a point operated 

 on by no other force flies forward and backward between fixed 



