Of KEPLER'S PROBLEM. 239 



Suppose x — tan -'; then cof v — : and, if we put/> 



to denote the perihelion diflance, =a (1 — s), and x — — 



5+v 



we fhall obtain, by fubflitution, 



- P( l + 0(' + **) - „ v 1 + *» 



? - (1 + + (1 — •) *» - ^ x 1 + x **' 



^ • 2 x 



Further, from the equation x zz tan -, we get, v — 

 Therefore, 



e ' (i + x*-y 



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

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

 have, from the nature of the curve, 



r= 2 P \ _ P_ 



1 -j- cof z g,i z 



cof - 



2 



Let y zz tan - : then — - — = i -f J 2 and z =: . 2 $ ., . rnn 



2 cof • * * + y 



2 



fequently, 



Now, equating the values of £ v and r 2 2;, that have juft been 

 obtained, and, omitting the common multiplier, 2 p* t there will 

 refult, 



and taking the fluents 



' + J 



It is manifefl, that this fluent requires no correction ; becaufe 



H h 2 : the 



