Of KEPLER'S PROBLEM. n$ 



power, or at leaft to an equation of the third power. It is 

 known, however, that in one particular iituation of the given 

 point, the problem becomes more fimple. This happens when 

 the given point is fituated any where in a flraight line, bifecting 

 the angle contained by the two lines given by pofition. In this 

 cafe, the problem becomes a plane problem, and leads to an 

 equation of the fecond degree only. 



Since, then, the ftraight lines AB, CD, interfecl one another 

 at right angles, the plane cafe of the problem will happen, when 

 the angle ACM is half a right angle ; that is, when n is equal 

 to an arch of 45 ° , or when the mean anomaly is a right angle. 

 This cafe deferves to be particularly confidered, on account of 

 the fimplicity of the folution it admits of. 



When n — 4c , it is obvious, that fin n zz cof n — : -=* ; and 



the general equation becomes 



_ 1 / i_ 1 \ I cof v — fin v m 



"72 X Um> co£»J "~^/2 X iinvcofv ' 

 fquaring both fides, 



1 cof 2 v -f- fin 2 v — 2 fin v cof v 1 1 — 2 fin u cof v 



£* — — x — — ^^— 1 ~ — v . 



2 fin ~v cof l v 2 fin z v cof 2 v ' 

 but fin 2» r 2 fin 1/ cof v ; therefore 



w 1 — fin 2v 



e % — 2 X — r • 



fin 2 2v 



Having reduced this formula, we fhall find 



fin 2 2v -f- - fin 2v zz — : whence fin 2v 1= + v/a ^ 2 + 1 1 



If we regard the problem, " Be inclinationibus" independently of 

 any application, and fuppofe that e may have all poilible de- 

 crees of magnitude, both thefe values of fin 2v may give folu- 



tions of the problem. One value, viz. fin 2v = ^ 2 e% ~*~ 1 * 



e % ' 



being always lefs than 1, will give a folution, whatever be the 

 magnitude of e ; it will even give two folutions ; becaufe fin 29, 



E e 2 being 



