206 THE ROYAL SOCIETY OF CANADA 



which may be further reduced to the usual forms 



1M>(^)! +(*!==,, etc.. etc. 

 A^ 52 C^ 



In particle dynamics, when the force is central, the differential 

 equation, in the case where the acceleration varies as the inverse 

 distance squared, is usually solved by converting 



(Pp uUp . ^ . f d-p dUp . . . 



— = — mto the torm — 7 = m > whence the mtesration 



df {TpY df dt 



proceeds. The following obvious treatment will appear more straight- 

 forward : 



26 



Starting with p = ra'^j3, we obtain 



d^-p jd'^r /ddW "^^^ Id/ de\ H-*^ 



dt' Ut~ \dt/ ) r dt\ du 



In central orbits, the acceleration perpendicular to the radius 



vector being zero, r"^ — =h, a constant. Whence — =hu", 

 dt dt 



dr _ du ^ _ _ d'^u (Pp _ /(Pu , \ V. 



dt de dp dd'' dp \de'' / 



for any central law of force. When the force varies as the inverse 

 distance squared this equals — /xM-Z7p, giving the differential equation 

 of the orbit, which is readily solved. 



