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



be a linear function of A 2 when each is regarded as a function of 

 u, v. This linear relation we can establish a priori ; for we have 



y+\ x = y/|W {&-<$) (1+A 2 ), 



Hence making %=§, we have 



(5a 2 -4cV 2 - (5c 2 -4« 2 )X 2 + 8(« 2 -c 2 ) =0. 



If we are content to leave the integral irrational in X or fju re- 

 spectively, then it presents itself under the form of a biquadratic 

 rational equation in u and v. Combining the above construction 

 of the integral with the well-known one through spherical trian- 

 gles, we obtain an interesting geometrical theorem, viz. that if 

 from a given spherical lune two arcs be cut off by an arc of con- 

 stant length, their sines may always be represented by the sum 

 and difference of the distances of two fixed points from a vari- 

 able point in a fixed straight line ; and moreover there will be 

 two systems of such line and associated points. 



Besides the general integral, we have also the singular ones 

 given by 



u=a or v = a, or u = c or v = c, 



indicating the familiar proposition that the product of the focal 

 distances from the tangents of an ellipse or hyperbola are con- 

 stant; u = a and v = c will correspond to an ellipse and hyper- 

 bola, of which the foci in the one are the vertices of the other, 

 and vice versa, If from any external point we draw a pair of 



tangents to either of these curves, -7-, i. e. -77 — --. and therefore 



if . dp' df-dg' 



Hr-, will have the same value at each point of contact ; so that if 



a, oJ and /3, /9' be the angles which the tangents respectively 

 make with the focal distances of the points of contact, we have 



7= 777, and also a! —a. the same in absolute magnitude 



cos ex! cos p' 



as /8'— ft, from which it is easy to infer ot = /3, a'=/3', showing 

 that the tangents to an ellipse or hyperbola make equal angles 

 with the focal distances at the points of contact, as is also known 

 from the theory of confocal conies. 



In precisely the same manner we may integrate the general 

 equation F (2p, 2q) =C, where 



