Of KEPLER'S PROBLEM. 209 



We have hitherto confidered the ferles of approximations to 

 the value of v to begin with the arch t ; but, in efFedl, the feries 

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



mula e =: e X ^^ —, we fubftitute n for v, the refulting value 



n — V 



of e will be e, to which the arch -x correfponds in the equation 



fin (« ■ — i) z:z e X fin v X cof v. It is clear, therefore, that the 



arch cr is derived from the arch », precifely in the fame way 



that 57 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 7i — »: taking the extreme cafe, when i zz. i, (in which 

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

 length of the arch « — c, when a maximum, is (Art. 3,) equal 

 to 7, correfponding to 28° 39' nearly. Therefore the arch n, 

 confidered as an approximation to v, is very wide of the truth : 

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

 feries, is neverthelefs inconfiderable, v;e fhall be entitled to con- 

 clude favourably with regard to the convergency of the feries. 



The error of the arch t is v — «-, 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 — 1*) — £ X fin r X cof v, 

 fin (« — ■r) z: e X fin 5r X cof 5r, 

 by means of which the arches v and •r are determined when the 

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

 and s X fin •3- X cof -x are evanefcent, when v zz o and «• rr o, and 

 alfo when v — go° and •r = 90° : therefore we fliall have 

 7; = ► r= T, not only when Ji n o, but alfo when « zz 90°. It 

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

 therefore the quantity v — ^ vaniflies, when n zz o and whert 

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

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

 K — ir will be a maximum. 



Since 



