304 MR. W. SPOTTISWOODE ON THE CONTACT OE CONICS WITH SURFACES. 
and if p be of the degree m in A, B, C, I), and V = ab A + /3B B + . . , it follows that 
Again, 
TD-d x <p = A V <P — map + rsrb' x p 
n/d^p = B V <p — mfip -\-™d' y p 
= — az^d x P 
( 39 ) 
whence 
= (t<7c) x — ct)nrd x p, 
vr 2 d x p =(AV —m-\-\a-\-^'d' x ){A^7 — ma-\-'aib' x )p 
nr‘ l 'd v 'd x <p = (B V — Vl-\- 1/3 + TO-Sy)( A V — ma A nrd' x )p 
(40) 
whence, again, 
t* 2 (M +l)^ x +..>={-( m + 1 (^ a +^ + '*) + ro (^+^y+-0K AV - Wa + ro ^) < P 
sr 2 (iPA+^y +• .)^= |~(m + 1 )l«+35i3 + . 0 + K$^ + £^+- •)K BV “ w / 3 +^) < P > 
and combining these expressions, 
ct 2 ($, . .)(B„ . .y<p=m(m + 1)(53, . .)(«> • .) 2< P — (2m+l)*r(8> • OK • -)K> • 0P+ OT *(&» • 0P» • OK 
Hence 
d x p 
3 Ah 
d x p 2 
A 
u 
d y P 
dy<Pl 
dy<P 2 
B 
V 
d z P 
d e P, 
d x p 2 
C 
w 
'btP 
d t p, 
d t p 2 
D 
k 
Ap 
Ap, 
Ap, 
. 
vH 
=to- - 4 — ma^ + ro-B x (J5 
— m/3<p+ro-By<p 
—mlp -}- <p 
/ m(m+l) (91, ..)(«> -OK \ 
I — (2m+l)®(3, . OK • 0(K • *)P ) 
'+ «*(& ) 2 P 1 
A 
B 
C 
D 
0 
u 
V 
w 
k 
vH, 
which, by a method precisely similar to that followed above in investigating the extra- 
neous factors, may be reduced to 
3'<P 
d x p , 
d' x p. 
A 
u 
d' y p, 
d'P* 
B 
V 
b'„<p 
d' z Pi 
d' s p 2 
C 
w 
d' t Pi 
d' t p 2 
D 
k 
Ap 
Ap, 
A<JJ a 
. 
»H, 
