454 ELEMENTARY ACTIONS. 



It follows, in fact, that the action exerted on the element b ought 

 not to change when a is replaced by c that is to say, ds by m*ds t 

 and r by mr; we get then, from equation (9), suppressing the 

 common factor ii'ds ds', and observing that the angles 6 and 6' are 

 equal, and the angle w is zero, 



cos 2 F (r) + sin 2 Of(r) = m* [cos 2 6 F (mr) + sin 2 Of(mr)~\. 



This condition should be satisfied, whatever be the particular 

 values of m, 0, and r; we must have separately 



m*f(mr)=f(r). 

 Making r=i, and m = r, we get 



r 2 F(r) = const = /*, , 

 r' 88 const M 



and, consequently, 



Thus the functions F(r) and /(r) are both inversely as the 

 square of the distance. 



The expression for the elementary action then becomes 



(u) d^= U * S -\ k cos# cos^' + sin^ sin 0' cos w , 



or 



(I2 ) 



J m >d ^ S> \(k - i) cos cos & + cos el . 



470. DETERMINATION OF THE RATIO OF THE Two CON- 

 STANTS. The last experiment (VI.) enables us to determine the 

 ratio k of the two constants. 



