THE CURVE ON ONE OF THE COORDINATE PLANES. 479 



Remembering 



and that 



we get 



where 



y 2 z 2 = s, y 2 + z 2 = r 2 , 



2/V + z 2 y* — (y 2 + z 2 )s = r 2 s 

 y 2 z* — z 2 y* = — (y 2 — z) 2 s , 



(III. ) ( r 2 X 2 + (b 2 - u 2 )(u 2 r 2 - 2¥)K 2 



0=S +2X2SXQXJ 

 ( + r 2 sX 1 2 



(IV.) ,X 2 j 



= (y 2 -z 2 ))-sX 1 2 C 



( +(6 2 -uVR' 2 ), 



X =u 2 [& 2 (Z 1 -w 2 )-<] 



u 4 



R' 2 = (t + L ) 2 - 8mu 6 s(i + L + L t ) . 

 Let us separate the terms affected by s from those which are not. Then we get 



(III.) r 2 X 2 + (b 2 - u 2 )(r 2 u 2 - 2¥)(t + L ) 2 



+s|-4X X 1 +r 2 X 1 2 



L + (b 2 - u 2 )(u 2 r 2 - 2¥)( - 8m)u%t + L + Lj) 

 (IV.) x 2 +u 2 (& 2 -u 2 )(* + L ) 2 



= 



^2 



+ s[ - X x 2 + u 2 (6 2 - u 2 )( - 8m)u 6 (i + L + L x )] 



this last neglecting the factor (y 2 — z 2 ), which gives a particular solution. 

 Designating the two equations respectively by 



A+sB = (Ill') 



C-sD-0 (IV) 



and remembering 



& 



we 



have 



r 2 u 2 = t, -m = ^g- 4> 



A = fu 2 [6 2 (^-u 2 )-i] 2 

 + t(b 2 -u 2 )(L + t) 2 

 -2¥(b 2 -u 2 )(L +t) 2 



B = t^(L 2 +t) 2 



+ (t - 2¥)(b 2 - ^){^){t + L + L X ) 



C = %*[&% - u 2 ) - tf + u 2 (b 2 - u 2 )(L + ty 

 B = ^(L 2 +t) 2 -^(b 2 - U 2 )(L +L 1 + t), 



VOL. XXXV. PART II. (NO. 13). 4 K 



