240 A NEW Md UNIVERSAL SOLUTION 



the two angles, v and x, are fuppofed to begin to flow together 

 at the perihelia of the curves. 



17. According to the fuppofition, the eccentricity e is nearly 



equal to tinity, and confequently, X n: — -;— will be a fmall 



fra(5lion : but, z.^ x 1^=- tan - j increafes from o to co, it is ob- 

 vious, that the fluent, obtained above, will be of ufe in com- 

 putation only to a certain limit, however fmall X may be fuppo- 

 fed to be. For the part of the fluent depending on X manifeftly 

 converges by the powers of the quantity Xx" ; and therefore, as 

 long as av" is a fmall fradion, fo long only can we compute x 

 when y is given, by means of the fluent : but when x has pafled 

 that limit, the fluent, in the form here given to it, is no longer 

 of any ufe in computation. 



But the fluent, although limited in its application by the 

 confideration jufl: explained, will enable us to compute x when 

 y is given, and to determine the angle v of true anomaly in- 

 the ellipfe, by means of the angle z in the parabola, for a con- 

 fiderable portion of the elliptic orbit lying adjacent to the peri- 

 helion, on either fide. We may therefore deduce from it a 

 feries that will ferve to compute the true place of a comet iii 

 the portion of its orbit which it defcribes during one apparition. 



In order to determine the angle v by means^ of the angle z, 

 we mull: firft find a value of x in terms of j : and, to avoid too 

 complex calculations, we Ihall neglecfl the terms multiplied' by 

 the powers of X, higher than the fquare. The extreme fmall- 

 nefs of X, in the orbits of all the comets, permits to Amplify the 

 calculation in this manner, and, neverthelefs, to obtain a refvilt, 

 that v'ill be fufliciently exaA duriag the time of one apparition. 

 Neglecting, then, the terms m;ukiplied by the powers of X 

 higher than the fquare, we have 



y f xl ^x , xl . ■, Trs x^l 



3 C 3J (3 SS is 73 



AflTume 



