THE CURVE ON ONE OF THE COORDINATE PLANES. 491 



§ XIII. 



We are now able to eliminate v and w from the expression of p. We have, 

 namely, 



We have also 

 hence with 

 we get 



We deduce 



tp-H = — iSi</> - H —j$j<j> ~ x i — k$k<p - x i ; 

 = iic+jv + kw, 



p = m -f i'v -f kw — iu — /-i — lc— — iv + kz . 





s =w— — 



Multiplying respectively by 



?iY', «Z', 

 and applying (I.), (II.), we get 



ziill = vajoTI — y^Z' = ?6 4 W 3 ' — y x Z'. 

 Having 



z x — 'Ea' 2 a 1 a. 2 , y 1 = 2,a 2 a 3 a 1 , 

 and 



u 2 = Sa 2 a x 2 , 

 we may write 



yicY' = Sa^j^W^a, — Y'a 2 ] 



zuZ' = S'a^fi^Wj'^ — Z'« 3 ] . 

 Of the factors between [ ], the first is 



~a 2 a*b 2 c 2 + b 2 a^b 2 c 2 + c 2 a x c^b 2 c 2 



Va?a x b 2 c 2 + b 2 a x b Y 2 b 2 c 2 + c 2 a 1 c 1 2 5 2 c 2 -i 

 2 L — a 2 a x 3 b 2 c 2 — b 2 b x z c 2 a 2 — c 2 c 1 3 a 2 b 2 J 



= a 2 [b 2 b 1 2 c 2 (a l b 2 — b x a 2 ) + c 2 c 2 b 2 ia x c 2 — a 2 c x )] , 



and the corresponding factor in zuZ' is the same with the index 2 changed 

 into 3, 



But 



a x \ - b x a 2 = SaiSfij — SfiiSaj = Safiji = — Syk = c 3 ; &c, &c. 

 The factors are respectively 



and 



. cc 3 [-bWe 2 +c\%b 2 l 



Ihis gives 



yuY' = Za^a^bW^ - c\ 2 b 2 b 3 ] 



zuZ' = 2a 2 a 1 a 3 [ - b 2 b 2 c 2 c 3 + c 2 c^b 2 b 3 ] . 



