TO SEVEBAL CENTRES. 203 



and whose investigation is less luminous than his great pre- 

 decessor's. Euler reduces his investigation to the integration 

 of the equation 



fj.dx vdy 



V x + of V y + y 8 ' 



and obtaining the relation between the angles made by the 

 two radii vectores with the axis. It is clear that Lagrange's 

 solution is obtained by another course altogether. 



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 varying in- 



versely 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 

 r 



only or to more centres than one. Herman, in the ' Pho- 

 ronomia,' turns away from it, merely observing that his 

 formula fails when m = 1. Clairaut, in his excellent com- 

 mentaries on the Trincipia,' his additions to Madame du 

 Chatelet's translation, deduces, chiefly from the Propositions 

 of the Second and Eighth Sections (lib. i.), a general dif- 

 ferential 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 therefore a polar one. 

 It involves the integration of j" Y dy. Consequently, when 



Y = , the case we are now considering, the integral con- 

 tains an unmanageable logarithm ; for the equation becomes 



He 



V(2B -logy 8 ) 



y) 

 makes no mention of this case, as, like Herman and most 



