THE CURVE ON ONE OF THE COORDINATE PLANES. 477 



we have 



L = 6 2 (u 2 +a 2 ). 



The equations (I.) (II.) contain the three elements y, z, and u 2 alone. 



To get rid of the three radicals R', Y, Z, we must have recourse to squaring twice 

 over. 



[(If we had, by a more direct process, operated at once on (1), (2), (3), in order to 

 get rid of the three radicals u, v, w, we would not have the advantage of knowing that 

 v and w were of the forms respectively 



E+C, E-C. 

 C being a radical)]. 



After squaring both members of (I.) we pass the radical YR' into the second member, 



the rational parts into the first, and we get 



16w 4 Yy-16uy ) = 8u 2 Yyb 2 K; 

 + ¥R' 2 -b\t + L Q ) 2 

 + 8 U yb%t + -L ) 

 or 



16uY(Y 2 — uV) ) = 8u 2 yb 2 YK , 



+ 8uyb%t+L ) 

 or by the values of Y 2 , R /2 : 



16%y ( - b 2 )[u 2 y 2 + b\u 2 - ¥) ) = 8u 2 yb 2 YK . 



-8mu 6 s(t+~L +~L 1 )b i 



+ 8uyb*(t+L ) 



We may suppress the factor 8u 2 yb 2 , by encroaching on s — y 2 z 2 . Thus putting 



Y' = - 2u\v?y 2 + b\u 2 - b 2 )] 



-muV6 2 (<+L +Li) 



+ u 2 (t+L ) 



we get 



where we put also 



As we have 



we replace, in Y r , 



Then as 



yY'=YK 

 zTl = -ZR'. 



Z' = - 2u\u 2 z 2 + b\u 2 - 6 2 )] 

 -mu^^+Lo+LJ 

 +u 2 (t + L ). 



t = u 2 y 2 +u 2 z 2 



u 2 y 2 by t — u 2 z 2 . 



L = b 2 (a 2 +u 2 ), 



Y' = u 2 r— 2t + 2u 2 z 2 — 2b\u 2 - 6 2 )~| 

 L+t+b 2 (a 2 +u 2 ) J 



-muV^+Lo+Li); 



