flri 



Of KEPLER'S PROBLEM. 221 



m — p' 



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



l(m-p') 



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

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

 we here denote by p, p\ p'\ &c. are manifeftly no other than the 

 doubles of the arches formerly denoted by w, t', *•", &c. re- 

 flectively. 



12. It is to be remarked, that, fince e is never greater than 1, 

 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 — 1, or fin <p — 1 when fin m — 1 ; and, consequently, 

 <p — 90 ° when m r= 90 °. Hence, it is eafy to infer, that the 

 arch <p 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 remark is neceffary to determine <p, when its line x 

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



When m — 90, or fin m == 1, we fhall manifeftly have the 

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

 rately. But though we have here fin <p — 1, and <p =: 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 ty become evanefcent. We may, 

 however, deduce the rule of calculation of Art. 10. from the ge- 

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



Resume the firft of the equations (A), writing ' <p, ^ and ?n, 

 for 2<p, 2^ and 2/7, according to the change made in the nota- 

 tion : viz. 



e fin <2> x cof ^ — 2 cof ^-^ — x fin -. 

 r 22 



Vol. V.— P. I. F f Suppofe 



