Of KEPLER'S PROBLEM. ' 207 



I,ET US now confider w, the firfl term in the Tenes of ap- 

 proximate arches ; this arch is the value of v in the equation 

 fin [n — v) -zz e X fm v X cof v, when s is fubflituted for e ; but, 



iince we have f = s X —, it is evident, that e is lefs than e, 



« V 



and confequently v will be greater than w. 



Again, take ■z', the fecond term in the feries of approximate 



arches. This arch is the value of v in the equation 



fin (« — li) ■=: e X {in ]/ X cof v, when i is fubflituted for e : Now, 



/ __ ^ fin (« — it) , fin (« — v) , ~ , , 



i — e X and e — iX — ^^ ; and fmce v has been 



fliewn to be greater than *, it is evident that n — t will be 

 greater than « — c : but the greater arch has to its fine the 



C f \ 



greater ratio ; therefore the fradion — — -^ '—, will be lefs than 



n — T 



the fraction ^ : confequently e will be lefs than e ; and 



therefore %' will be greater than v. 



^ .- . ^ r 1 fin (« — v) 

 And, in general, if, in the formula e ■=. iX '^^i;: , we fub- 



ftitute a greater arch for v, we fliall have a value of e greater 

 than its true value ; but, if we fubflitute a lefs arch for c, we 

 fliall have a value of e lefs than its true value : but we have de- 

 monftrated, that, in the equation fin (n — f) — ^ fin c X cof i/, 

 the greater e is, the lefs will the arch v be : from which confider- 

 ations the truth of what we have afTerted above is evident, viz. 

 that the arches r, •t', ^r", &c. continued indefinitely, are alter- 

 nately too fmall and too great. 



7. Let us now compare together the alternate terms in the 

 feries of approximate arches ; and it will not be difficult to per- 

 ceive, that the firft, third, fifth, &c. terms, which have been 

 fhewn to be all lefs than half the arch of eccentric anomaly, 

 continually increafe ; but that the fecond, fourth, fixth, &c. 

 terms, which have been fhewn to be all greaterthan half the 

 arcTi of eccentric anomaly, continually decreafe, 



Dd2 Foa 



