204 ON FORCES OF ATTRACTION 



others, he seems unwilling to approach it; undoubtedly, 

 however, such is the application of his formula. 



Keil, in his paper on Central Forces in ' Phil. Trans.' 1708, 



p. 174, gives the case of the force as and reduces it to 



finding = P, the perpendicular to the tangent. 



(6 - log r)* 

 By one process grounded on Prop. XLL, Lib. I., this 



result is obtained for the case of , that is 



r , 2 2 y|- 



C(*2dxd*x-\-1dyd*y ' f J~^~ (x d x -f- y d y^ 



\ j~a = ^ A 1 9~. 5 or 



a r or 4- y 



*J u tJ ' n 



C* dx* A- dip 



j~^ 2 log O 2 -f- f) - c = ; and d t* being 



(y dx xdyf (* 



= 2 , the equation becomes 



(ydx - xdy}* : 

 - lo < 



hr 

 The process grounded on the formula / = =5 is, if 



l } It 



possible, more hopeless ; for this gives 



ft(y 4. (x - C JY X (dxtfy - dytfx) _ _ 1 _ 



2 (ydx - xdy + c dy}* ~ ^ + (^ _ C ))T* 



or h (y 8 + (# c) 2 ) (dxd*y dy d* x} = 2 (ydx xdy -\- 

 cduY or kdy - ^(ydx-xdy + cdy}* 

 ~d^ ~ J dx*(f + (x-c?) 



The difficulty follows wherever that proportion enters 



into the investigation. Thus in the problems connected with 

 different centres, when it is found that forces varying as 



and , being combined with forces varying as the dis- 



