2oa A NEW and UNIVERSAL SOLUTION 



Foe, it is obvious, that the greatefl value of c, in the formu- 

 la, e zz: I X - — , is when /; — i* z= o, in which cafe i" — e : 



n — V 



therefore, the arch «• will be the leafl of all the approximate 

 arches, x, «■', t", &c. : therefore •z is lefs than ^r", the third 



term in the fcries. Again, we have s ir s X ^ ~, and 



H T 



i" zz £ X 77—' : and fince c is lefs than ir", it is manifeft 



that }i — c- wilf be greater than /; — -z" : confequently e' will 

 be lefs than i" ; whence it follows, that -x , the value of v corre- 

 fponding to i, will be greater than t'", the value of v correfpond- 

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

 the ferics y, ff, -tt", %'\ &c. is lefs than the third term ; but that 

 the fecond term is greater than tlie 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 demonflrated, that the arches in the 

 feries ff, ■?/, t", &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 

 ff, i:', 5r", &c. form a feries of approximations, converging to the 

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

 alternately in defedl and in excefs. 



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

 of the feries t, it, ^r', &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- 

 iution is to be reckoned more or lefs perfe(5l and valuable. 



We 



