Prof. Sylvester on Periodical Changes of Orbit, fyc. 293 



Then 



{c 2 -v 2 ) 2 du 2 -(u 2 -c 2 ) 2 dv 2 =={c 2 --a 2 ){c 2 -v 2 )dii 2 - (c 2 -« 2 ) (w 2 -c 2 )^ 2 , 



or 



du 2 (dv) 2 



(« 2 -a 2 ) (u 2 -c 2 ) {v 2 -a 2 ){v 2 -c 2 ) *~ 



The integral, therefore, of this equation must express the fact 

 that u and v are, or may be regarded as, the sum and difference 

 of the distances of two fixed points distant c apart from any point 

 in a fixed straight line, the product of whose distances from those 

 points is c 2 — a 2 , or also, if we please, as the sum and difference 

 of the distances of two fixed points distant a apart from any point 

 in a fixed straight line the product of whose distances from the 

 points is a 2 — c 2 . 



Thus, parting from the first construction, if we write y + \x=~L 

 as the equation to the straight line, the origin being taken mid- 

 way between the two points, and tbe axis of x coincident with 

 the line joining them, we obtain 



c 2 



or 



L 2 =J\ 2 +(c 2 -fl 2 )(l+X 2 ); 



we have also 



v .a o o . ° 



u 2 =y 2 +-- — cx + x 2 ; v 2 =y 2 +- +cx + x 2 ; 

 so that 



2\2 



f- 



_ a c 2 _ (v 2 -u 2 ) 

 2c ' y -" V 2 2c 2 



and that the required integral will be 



v 



. 2 - c 2 (v 2 -u 2 ) 2 ^v 2 -u 2 



+ \/^+ (c 2 -* 2 ) (i+x 2 ) =0, 



which, completely rationalized, will lead to an equation of the 

 eighth degree in u, v, and quadratic in X 2 . 



Ji. similar rational equation in u, v, /u, 2 can be obtained by in- 

 terchanging a and c with one another, and \ with /jl ; and as 

 each equation represents the complete integral, y? will necessarily 



Phil Mag. S. 4. Vol. 31. No. 209. April 1866. X 



