THE CURVE ON ONE OF THE COOEDINATE PLANES. 



485 



and 

 Hence 



lb Cv-i (A/a (X/n 



vw 



u*M,' 



a, 



- = the same expression than = . 



1 a, 



In the case of M/' we know that 



which, as 



^K = 



rJ,K= -ifM^ + M^Q +M" 1 R 2 ] 

 + fc[M 1 P 2 +M' 1 R 1 + M" 1 Q], 



gives zero identically for the coefficients of j and k We have then by the second of 

 these identities 



m» n ( M i\r x) , t> i M'j Mj 



or 



M/'w 3 



M~t) 



\ m w J 



u° 



Now the coefficient of — is 



w 



a^ x — a 2 y 1 = zero. As — Oj X z x = — na^a 2 , 

 we have 



The expression in [ ] is 



u\b 2 - w 2 ) + 6 2 (v 2 - a 2 - 6 2 ) + w 4 

 = b 2 [u 2 + v 2 -a 2 -b 2 ] = b 2 (b 2 -w 2 ) 



We have then 



But 

 hence 



= — b 2 na 2 . 



M /M\ 



M" x w 3 Q = — I . a 2 b 2 na 3 2 = ( — l )a 3 & 2 a;i . 



(Xj \ a l/ 



% 3 Q=6 2 a;i ; 

 a 3 — a x ' 



Thus in the case b 2 = c 2 , the three expressions of Mi, M/, M/', when equated to zero, 

 give only one distinct equation. 



We may remark that M/ and M/' may be annulled by putting 



P 1 = 0,P 2 = 0. 



These two equations give four solutions as to a 1} a 2 , a 3 , of which one is also given by 



