Mechanical Equation to the Motion of Material Points. 331 



their mutual distance remains constant. Then we need not take 

 the mean value of <f>(r), but can write 



*W=/W (49) 



Further, for this case u also is constant, and equation (46) 

 changes into 



\==z\/V 2 ==m. 



The product iu here occurring has a simple signification ; for it 

 is the relative length of the path — that is, the length we obtain 

 when we conceive one of the points at rest and ascribe the velo- 

 city u to the other. This path is a circle with the radius r ; and 

 hence we obtain 



\=iu = 27rr, 

 and consequently 



A, 



This value inserted in equation (49) gives 



*G&=/Mi ( 5 °) 



and by this the form of the function f(X) is determined. If, as 

 before, we introduce p with the signification 



*-£ < 51 > 



there results 



and by employing this equation, (48) is transformed into 



W) = $(P)- • • (52) 



Returning now to equation (47), according to what precedes 

 we can write it in the following form : — 



^( /9 )=-^ 2 -Sl0g(27T / 0), 



or 



T ' ' m-\-m l p 

 from which follows : — 



u 2 =(m + m l )p<j) l (p) (53) 



"When, further, in equation (46), in the place of X we put the 

 product 27T/9, there comes 



2irp = iJ^ } 



