Prof. Sylvester on Periodical Changes of Orbit, fyc. 297 



and if the areas of the triangles GHK, HKF, KFG, FGH be 

 called V v G v H 2 , K 1 respectively, and P be any point in space, 

 it is easy to prove that 



Y l . PF 2 - G x . PG 2 4- H, . PH^Kj . PK 2 =E, 



■where E is a sort of geometrical invariant independent of the 

 position of P. Its value may be expressed by the equation 



FG 2 FH 2 FK 2 



GF 2 GH 2 GK 2 



HF 2 HG 2 HK 2 



KF 2 KG 2 KH 2 



By making P coincide with F we find 



±E = FG 2 .HKF + FH 2 .GKF-FK 2 .GFH. 



Hence, if the position of K be determined by linear coordinates, 

 x, y, and of F, G, H by coordinates of the like kind, it is obvious 

 that E becomes a rational quadratic function of x, y ; F 1? Gj, 

 H 1 linear functions of x, y\ and K 1 independent of x, y. 



Let P be any point in a twisted Cartesian whose foci are 

 F, G 3 H ; p, a, t the distances of P from these foci. Then we 



-16E 2 = 



Z/)+m(7 + 7lT+j0 = O, (1) 



/'p + mV + w'T+y = 0, (2) 



where I, m } n, p ; /', m', n 1 , p ! are constants. 

 Let v be the distance of P from K, then 



F 1 p 2 -G 1 <7 2 -fH 1 T 2 -E=-K 1 i; 2 , ... (3) 



and v will be a linear function of p, a, t, provided that the values 

 of p, cr in terms of t determined from (1) and (2) make the 

 left-hand side of (3) a perfect square. 

 The condition that this may happen is 



J?,3 



0; 



} 



0; 



l; 



I' 



;- 



-Gi; 



0; 



0; 



m; 



m! 



; 



0; 



H l5 



0; 



n; 



n 1 



; 



0; 



0; 



-E; 



Pi 



p> 



l ; 



m; 



n ; 



Pi 



0; 







l' ; 



m'; 



n<; 



p'; 



0; 







= 0. 



(4) 



It is easy to see that the determinant above written consists 

 exclusively of terms in which only binary combinations of F„ G lf 

 Hjl, E appear. Consequently equation (4) is an equation of 

 the third degree in x, y. When this equation is satisfied, K is 

 a focus just like F ; G, H. Hence we may conclude that any 



