ME. W. SPOTTIS VYOODE ON THE CONTACT OF CONICS WITH SUEEACES. 295 
Further, if we agree upon the proper mode of development we may write 
• 
B 
C 
DB;V + .. 
■ + 2. 
A 
c 
L>W + 
B 
id 
nd 
A 
Ul 
v' 
1 1 
C 
id 
w 1 
vd 
B 
w' 
id 
nd 
D 
mi 
n' 
h 
D 
d 
id 
h 
u l 
w' 
v' 
1 ! 
A 
dx V= 
=AY suppose 
w' 
«i 
vd 
nd 
B 
v' 
id 
w, 
id 
C 
cL 
l 1 
nd 
id 
*, 
D 
A 
B 
C 
D 
. 
. 
^x 
3, 
B, 
• 
* 5 
which expression, by giving obvious values to 91, . . Jf, . ., may be written thus : 
5?3;V+. .+24ra,3,V+. .=(3, 93, €, S, f, ©, % 1, HI 3,)’V. . (IS) 
Lastly, putting 
H =u, 
w' 
v' 
t 
A 
w' 
Vi 
id 
nd 
B 
. v' 
id 
Wi 
id 
C 
V 
nd 
id 
*i 
D 
A 
B 
C 
D 
• 5 
(19) 
the expression for n 2 V finally becomes 
Av-(i+ ? ~r)fiH=o, 
and consequently the system Y = 0, □Y=0, n 2 V=0 may be replaced by 
(\ + MV' (20) 
u v w k \ n — l / H * ' 
If in the foregoing expressions we put a’ = 0, /3' = 0, y' = 0, S' = l, § = 0, ^ = 0, we shall 
have the case of plane curves, and as the last suppositions give w' = 0, A = 0, B = 0, 
C=0, D=ux-\-fiy-\-yz, the expression (20) then reduces itself, as it should, to that given 
in equation (16) of the memoir above quoted. 
§ 3. Elimination of the Constants of the Quadric V. 
Before proceeding to the application of the formulae (20) to the present problem, it 
will be convenient to premise that if <p, be any two rational integral and homogeneous 
functions of x, y , z, t, the nature of the operation A is such that 
'J'Ap + pA\J/-|-2(S, B, C, D, Jf, (3, fy, 2C, iH, $,)(d a p, d x <p, d t <p)(d u ,\}/, d^, d £ yp, d,\p); 
(21) 
