Prof. Sylvester's Astronomical Prolusions, 6£ 



we have to satisfy the equation 



^ -y-cose+o TT COS2' + V + V'=0. 

 Writing 0= —7= = -j=, and taking ^(s), any arbitrary func- 



tion of g, we may write 





or 



dV * 2 TT 4c? 



jrjp'j 



£fo ' fl 8 -C 8 + ^ 8 a* — (* + CZ* 



and then 



lb + a*-<f + yz* ~ a 8 + y 8 -7* 9 W ' 

 Integrating these equations, we find 



also 



Hence, making 



1^ = 



2*(a 2 -c 2 4-c* 2 ) 

 we have 



/ ( 7 - c + **)« (7-^ + ^ 2 ) 3 ' 



(c-7+?) 8 " f (c-7H-^ 





</> being any arbitrary function, and 2 representing —-=. or —7=. 



A similar method will apply to the determination of the forces at 

 the foci whereby any conic may be described*. 



* Employing the equation 



tr iwil (dV , cTV'\ . n 



rr' 

 replacing p cos i by its equivalent — , writing r — a = a—r'=z, and decom- 



