2c8 A NEW and UNIVERSAL SOLUTION 



For 3 it is obvious, that the greateft value of e, in the formu- 



, „ fin (n — v) i i • i r 



la, e — g X t -, is when n — » — o, in which cafe e r= £ : 



n — v 



therefore, the arch * will be the lead of all the approximate 

 arches, *■, sr, *■", &c. : therefore ?r is lefs than w", the third 



term in the feries. Again, we have i — z X m v ff ^j an( j 



/• f //x 



g" — g X 77—' : and fince * is lefs than sr", it is manifeft 



that » — k wilf be greater than n — v" : confequently ( will 

 be lefs than i" ; whence it follows, that t', the value of v corre- 

 fponding to g, will be greater than <x"\ the value of v correfpond- 

 ing to i". Thvis, then, we have fhewn, that the firft term in 

 the feries t, t , t", t", &c. is lefs than the third term ; but that 

 the fecond term is greater than die fourth : and, it is clear that, 

 by a fimilar mode of reafoning, we may prove that the third 

 term is lefs than the fifth ; but the fourth term greater than the 

 fixth ; and fo on. 



8. It has now been demonftrated, that the arches in the 

 feries ?r, 5/, *■", &c. are alternately lefs and greater than half the 

 arch of eccentric anomaly ; and further, that the terms in the 

 feries, lefs than that arch, continually increafe, while the terms 

 in the feries, greater than that arch, continually decreafe ; 

 whence, it is manifeft, that by computing more and more terms 

 of the feries, we fhall have the value of the arch of eccentric 

 anomaly within narrower and narrower limits. The arches 

 t, a-', -r", &c. form a feries of approximations, converging to the 

 true length of half the arch of eccentric anomaly, and erring 

 alternately in defecl and in excefs. 



And here a queftion occurs. It may be alked, Do the terms 

 of the feries t, ic ' , w\ &c. converge flowly to the arch fought ? 

 or, Do they converge rapidly to it ? According to the anfwer 

 that we fhall be able to give to this queftion, our method of fo- 

 lution is to be reckoned more or lefs perfect and valuable. 



We 



