. '■Of KEPLER'S PROBLEM. 239 



Suppose x = ta'n'^'; then cof -z^ = : and, if we pntp 



to denote the perihelion diftance, =a (i — s), and a — ^ , y 

 we fhall obtain, by fubftitution, 



^ ~ (I + + (I — ^' - I + X ;«;-• 



Further, from the equation x = tan -, we get, v — ■ » : 

 Therefore, 



Again, the perihelion diftance of the parabola being, by the 

 fuppofition, equal to the perihelion diftance of the ellipfe, we 

 have, from the nature of the curve, 



r- ^P - P ^ 

 I + cof Z -I z' 



2 



Let V = tan 5 -. then — — = t + j« and z = ^-^ , 



2 



con- 



fequently, 



r^ z z: a/*' X i (l H-j'). 



Now, equating the values of f* v and r' z, that have juft been 

 obtained, and, omitting the common multiplier, 2 />', there will 

 refult, 



and taking the fluents 



It is manifeft, that this fluent requires no correction j becaufe 



H h 2 the 



