Prof. Sylvester's Astronomical Prolusions. 



59 



perfectly account a priori for the existence of the excluded infi- 

 nite values of e 2 . 



Since FP±GP=±2« and FQ±GQ=±2a, 



G will be any point in the intersection of either of two circles 

 C v C 2 with either of the two K 1? K 2 , where the squared radii 

 of C„C 2 are (2« + FP) 2 , and of K v K 2 (2« + FQ) 2 , (2«-FQ) 2 

 respectively. Consequently there are eight real or imaginary 

 positions of G at a finite, and eight at an infinite distance. 



It is obvious that, if we restrict the orbit to being elliptical, 

 there can never be more than two admissible solutions; but 

 treating the question more generally, any even number of solu- 

 tions whatever may exist from to 8, both inclusive. I am in- 

 debted to my able friend Dr. Hirst for the following figure, 

 illustrating the interesting case where all eight solutions are real 

 and hyperbolas. 



In this figure 



p(=FP)>2a, p'(=FQ)>2a, 



and likewise 



p+p'—4a>c, 



4a+f)— p' -<c. 



Supposing F P, F Q to be each greater than 2a, there will be 

 no difficulty in verifying the following statement. 



One pair of hyperbolae, in which P, Q lie in the branch con- 

 taining F, will be always real ; a second p air, in which they lie 



