41] MOTION OF PLANET. 143 



The first of these is an ordinary differential equation for E giving 



The equation 140) is to be treated in a similar manner writing 



142) jF =(#) + $O). 



Proceeding as before, multiplying by sin 2 # and transposing, 



143) S i^ 2 - 2/3si ^ = 

 From which, as before, we must have 



whose integrals are = 

 146) e 



Substituting the values of H } <9, Ct), 



147) A = f\/ -* + 2h.dr+ A/2/3 - -^ d& + Vy w. 

 J \ r r* J Y sin-# 



This solution contains the three arbitrary constants, /3, y, /&. Differ- 

 entiating by them we obtain the integrals 



dA_ C dr C d%> 



^ = /, ! y" 2 - 2 l+^ jT/tT-% 



*s V r r z *s f sm 2 



dA _ r d^ y _ f 



148 ) ^ ~ / ^T/2^8in 4 ^-ysm 2 ^ 2 V7 ~~ ^ ; 



a^i /* dr 



Th= h 



V"- 



If we put y = 0, necessitating according to the second equation 

 9 = 0, the first equation becomes 



149) v ' ' ^' 



the equation of the path, which, on performing the integration 

 indicated, takes the form obtained in 20 equation 23). 



