292 MR. W. SPOTTISWOODE ON THE CONTxlCT OE CONICS WITH SURFACES. 
( 10 ) 
and writing, further, 
ax -\-(3y -f y =to ' 
a'x -f /39/ -f y'z -f It' = to-' 
ara-' — «to' = A 
/3'to — /3to' = B 
7 'ra- — yro-' = C 
cS’to — ^to' = D, 
the quantities «, b, . . A, B, . . will be found to satisfy the following relations, useful in 
subsequent transformations, viz. : — 
bz — cy—ft =A 
ex — az — gt =B 
ay — bx—ht =C 
fo+gy+te 
Aa'-f B_y +C£+D£=0 
BA-Cy+«D = 0 
C/_AA+6D=0 
A ? -B/+cD=0 
Aa-(-B54-cC =0 
af -\-bg -\-ch =0 
toB i3 ,A = «A — «A, to^A = kB — (3A, toS x ,A = «C — yA, nrB^\=:aD — A4., 
toB,B=/3A— «B, to^B=/3B-/3B, to^B=/ 3C- 7 B, ®B 4 B=/3D-iB, 
to^C = 7 A-«C, toc^C = 7 B— /3C, ®^C= 7 C- 7 C, toB,C= 7 D-SC, 
tocUD=AA — aD, toB^D — 3 B — (3D, to( 3.1) =o C — yD, — dD. 
And in terms of this notation the developed form of □ will be given by 
— □ =(yh— wg-{-ka)d x -\-{wf— uh-\-kb]b a -\-(ug— vf-\-kc)b~-\-(— ua— vb—wcjd t . (12) 
( 11 ) 
This being premised, our first object is to investigate, as was done in the case of 
plane curves, an expression for CUV, which in virtue of (12) will consist of two parts: 
first, terms of the form ( hDv — yDf-)-«Dj!:)B„ ..; and secondly, terms of the form 
(vh— wg-\-fcayd% . . Referring to (12), we have 
hOv— g[3iv-\-a£3k= a, a', u, hw' —gv' -\-aV 
(3, (3', v, hv , —gu'~\-am! 
y, y , iv, hv! —gw ,-f an' 
S, l' , k, Jim ' — gii!-\-ak l 
