THE CURVE ON ONE OF THE COORDINATE PLANES. 497 



Grouping the terms independent of B , B' separately from those which are 

 dependent on B , B', and putting 



4Y' = X +X 



4Z' = X -X\ 

 we have (remarking 2c c' = C), 



X = « 2 a %' 2 C 



+ fra' i a C'c 2 



+ cV% C'c' 2 

 X = a' 2 a G'[a 2 a 2 + a'\b 2 c 2 + cW>)] 

 X =-2W 1 'u?, 



and 



Replacing 

 we get 



Putting 

 we have 



X'=-a% 3 W 



+5VV[B a C4-2BV] 



+ cW 2 [B « C'-2B'c' 2 ]. 



W' = B C'(l + a 2 ) + 2B'a C 



X' = B a C'[-«V(l + a 2 ) + a\b 2 c ( 2 + cV 2 )] 

 + 2 B'[ - a 2 a 4 C + a'% b V - cV 4 )] . 



b 2 c 2 + c 2 c 2 = d 



X = a C'[a 2 a 2 a"* + a*d] 

 X' = B a C'[ - a 2 a 2 ( 1 + a 2 ) + a'*d\ 

 + 2B'[-aV0 + a'%]. 



§ XVL 

 The equations (I.) (II.) of § XII. put under the form 



»Y'-« 3 W 8 '=0 



wZ'- ? 4 3 W 3 ' = 



when rendered rational as to u, v, w, and when multiplied by 2 5 , are of course 



(I. )' (2&) x (2 4 Y' 2 ) - u 6 . 2 5 W 2 ' 2 = 



(II.)' (2vfi) x (2 4 Z' 2 ) - vfi. 2 5 W 3 ' 2 = . 

 The first becomes 



(x +x')(X + X') 2 - tt 6 2a ,2 b 2 (W -W7 = 0. 



We group the terms so as to put those of even order in B , B' together, and 

 those of uneven order together, replacing 2& 2 by I +B . 



VOL. XXXIII. PART II. 4 D 



