fin 



Of KEPLER'S PROBLEM. 221 



m — />' 



i X -, &c. ; the feries p, p', p", &c. will converge very 



L{m-p') 



quickly to the exadl value of the arch of eccentric anomaly, 

 erring alternately in defedl and in excefs. For the arches that 

 we here denote by p, p\ />", &c. are nianifeftly no other than the 

 doubles of the arches formerly denoted by v, % , x", &c. re- 

 fpe<5lively. 



12. It is to be remarked, that, fince e is never greater than i, 

 _the cubic equation in the rule above, is of the form which ad- 

 mits only one real root, fo that it may either be refolved by 

 Cardan's rule, or by the ordinary methods of approximation; 

 its root is always pofitive. 



It is to be remarked too, that in the fame cubic equation, 

 X •=. \, or fin <p zi I when fin wz i:: i ; and, confequently, 

 <p rr 90° when m ■=. 90". Hence, it is eafy to infer, that the 

 arch ip is lefs, or greater than a quadrant, or equal to it, accord- 

 ing as the arch m is lefs, or greater than a quadrant, or equal 

 to it. This remai-k is neceflary to determine (p, when its fine x 

 is given, on account of the ambiguity of the fines. 



When in -zz 90, or fin 7/1 — x, we fhall manifeftly have the 

 cafe of the problem that we before (Art. to.) confidered fepa- 

 rately. But though we have here fin (p ::= i, and cp =: 90°, we 

 fhall in vain look for a folution of this cafe from the general 

 rule, (Art. 11.) : becaufe the firft and third terms in the pro- 

 portion for computing cof <J/ become evanefcent. We may, 

 however, deduce the rule of calculation of Art. i o. fi-oni the ge- 

 neral inveftigation of Art. 11. in the following manner : 



■Resume the firft of the equations (A), writing tp, '^ and m, 

 for 2(p, 2<l' and 2», according to the change made in the nota- 

 tion : viz. 



e fin (p X cof 'I' — 2 cof ?-^ — x fin -. 



2 2 



VoL.V.— P. J. Ff Snppofe 



