58 Prof. Sylvester's Astronomical Prolusions. 



pression 



™- 2(cW) 



cro' + ^t */(c 2 -<r 2 )(c 2 -<7' 2 )' 



where o-' + <r= +4<z. 



Since we have also (calling i the angle between c and a) 



O" s 



cos *= — in the first case, and cos i = — in the second case, the 

 ec ec 



problem of determining the conic, of which one focus, the major 

 axis, and two points are given, is thus completely solved. This 

 of course comprehends the analytical solution of the problem of de- 

 termining the magnitude and position of the orbit of a planet from 

 the periodic time, two heliocentric distances, and the included 

 angle, of which no mention is to be found in any of the ordinary 

 books of astronomy, or even in the Theoria Motus, where one 

 would naturally expect to find it. 



There are thus eight values of e 2 , and the solution is eight- 

 fold. The sign of cos i being left ambiguous does not raise the 

 number to 16 ; for this ambiguity depends upon the fact of the 

 direction of c being incapable of analytical representation ; only 

 one of these values of cos i will appertain to any stated case. If 

 F be the given focus, P, Q the two given points, and G the second 

 focus, by rotating the figure about the line FG P, Q come into the 

 positions P', Q!; c, s } a remain unaltered ; but the angles between 

 Q'P', FG, and between QP, FG become supplementary. If we 

 chose to effect a direct solution of the problem of determining the 

 orbit without the aid of the eccentric anomalies, we should have 

 to eliminate 0, 6' between the equations 



This elimination will be found to lead to a quadratic equation in 

 e 2 , the coefficients of e 6 and e B vanishing; and we thus obtain an 

 eightfold solution as before, but in a more involved form. Or, 

 again, we might write 



(a(l-e 2 )+^) 2 =p 2 , 



(«(14-e 2 )W) 2 =/>' 2 , 



xri +yy l -=.c*, 



rf + yV^p*, z'*+y'*=p'% 



and between these five equations eliminate oc, x l , y, y\ By the 

 general theory of elimination, e 2 should rise to the sixteenth 

 degree in the resultant; but in fact it will rise only to the 

 eighth. The following obvious geometrical construction will 



