330 Prof. R. Clausius on the Application of a 



only under their reciprocal action, in closed paths, about each 

 other, their common centre of gravity must remain fixed, aud 

 hence 



dx dx, 



by which the preceding equation is transformed into 



(dx dx{\* , T fdx\ 2 , fdxXl 



mm {di-"dt) = ^ + w '> Hdt) +m {lt) J • 



Exactly similar equations are valid for the y and z directions ; 

 and if we suppose these three equations added, we obtain 



mm 1 u' 2 = [m + mj (mv 2 -f- w^gjv 

 or 



mv^-+-miV*= -u? (43) 



If this value of rmfi + m-p^ be introduced into equation (41), and 

 then the product mm 1 be taken away, the result will be : — 



^)=^ i d^+irnhgi). . . . (44) 



For still further abbreviation we will write this equation in 

 the following form : — 



8#T=^Slog(*VP} (45) 



And here we will again introduce a single symbol for the pro- 

 duct which stands under that of the logarithm, putting 



X=tV^ (46) 



by which we obtain 



^)=: _1__ SbgA, (47) 



This equation can now be treated just as equation (5) was. 

 The left-hand side being a variation, the right-hand side must 

 be so too; and hence u 2 must be a function of X; and thence 

 it follows further that <b(r) must also be a function of \. Hence 

 we put preliminarily 



W)=/W> ( 48 ) 



and put to ourselves the question whether, for any special kind 

 of motion, the form of the function f(X) can be found. This 

 can be done when the two points so move about each other that 



