APPENDIX. 421 



f* 



Q 1 ^/ ~ i , 



V- - 



The integral of this, calling C + p = L 2 , is cos r 



= Q , or - = 2* = p + L cos a, the general equa 

 tion to a conic section. 



. NO. III. FORCE VARYING INVERSELY AS THE 

 DISTANCE. 



IT is remarkable that what at first sight seems to be the 

 most simple of all the cases, that of the central force vary 

 ing inversely as the distance, or of m = 1 in ~, should be 



found so much the most difficult of solution, and that, 

 whether the proportion of - enters into motion related to 



one centre only or to more centres than one. Herman, 

 in the Phoronomia, turns away from it, merely observing 

 that his formula fails when m = 1 . Clairaut, in his excel 

 lent commentaries on the Principia, his additions to Madame 

 du Chatelet s translation, deduces, chiefly from the Pro 

 positions of the Second and Eighth Sections (Lib. L), a 

 general differential equation for the curve described by a 

 body under the influence of a centripetal force as Y, a 

 function of the radius vector ; and the equation is there 

 fore a polar one. It involves the integration of f Y dy. 



Consequently, when Y = - , the case we are now con- 



/ 



sideriug, the integral contains an unmanageable logarithm ; 



E E 3 



