Of KEPLER'S PROBLEM. 209 



We have hitherto confidered the feries of approximations to 

 the value of v to begin with the arch r ; but, in effect, the feries 

 may be confidered to begin with the arch n. For, if in the for- 

 mula e — £ X - —, we iubftitute n for j>, the refulting value 



n — v 



of e will be £, to which the arch t correfponds in the equation 

 fin (« < — v) zz e X fin v X cof v. It is clear, therefore, that the 

 arch % is derived from the arch /?, precifely in the fame way 

 that v is derived from <x ; or that any other term in the feries 

 is derived from the term that immediately precedes it. 



The error of the arch n, confidered as an approximation to 

 v, is n — n: taking the extreme cafe, when % zz 1, (in which 

 cafe the convergency of the feries is evidently the flowed), the 

 length of the arch n — v t when a maximum, is (Art. 3.) equal 

 to ~, correfponding to 28 ° 39' nearly. Therefore the arch n, 

 confidered as an approximation to i> 9 is very wide of the truth : 

 And, if we can prove that the error of r, the fecond term in the 

 feries, is neverthelefs inconfiderable, we mall be entitled to con- 

 clude favourably with regard to the convergency of the feries. 



The error of the arch % is v — ir y and we are now to in- 

 quire, to what degree of magnitude this arch may attain. For 

 this purpofe let us confider the two equations, 

 [n — v) zz g X fin v X cof y, 

 fin {n — sr) - j X fm ^ X cof w, 

 by means of which the arches v and % are determined when the 

 arch 11 is given. It is clear, that the quantities s X fin v X cof v 

 and s X fin t X cof tt are evanefcent, when v — o and v zz o, and 

 alfo when ^ r 90 and t zz go° : therefore we ihall have 

 n — v zz v, not only when 11 — o, but alfo when 11 zz 90 °. It 

 has alfo been fliewn, that the arch v is greater than the arch t : 

 therefore the quantity » — t vanifhes, when n zz o and when 

 n zz 90 °, and between thefe two limits it is always pofitive ; 

 confequently there is an intermediate value of tf, where the arch 

 v — % will be a maximum. 



Since 



