302 MR. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES, 
we have (remembering that H = 0) 
x y z D£ Jet 
(aa.+ ..)P ®b,+ ..)p («&.+ ..)? d(«b w +..)p *(»,+ .. )P; 
which vanishes identically. Hence H is a factor of (36) ; and on dividing it out the 
degree of the expression (36) is reduced to Ibn— 20. 
It remains now to be shown that u is likewise a factor of (36). Putting u= 0, that 
equation becomes 
Uy 
X- 
-xd x P + P 
. A 
u 
w' 
X- 
-xd y P 
• . 
. B 
u 
V 1 
X- 
■xd x P 
. . 
. C 
w 
V 
X- 
#chP 
13 
V 
A uX- 
-a?AP + 2HA t X- 
-2(ad,+..)p . 
. 0 
Hi 
1 
X 
. . 0 
0 
Uy 
x~d_ 
r P+P 
. . A 
u 
w' 
P 
. . B 
V 
v' 
x~b z 
P 
. . C 
w 
l 1 
xd t 
P 
. . 13 
p 
An zAp + 2Hd 2 X-2(a^+. .)P • • 0 HI; 
and following a process similar to that adopted in the former transformation, this may 
be reduced to 
1 
X 
. . 0 
0 
Uy 
P 
. . A 
u 
w' 
• 
. . B 
V 
V 1 
. . C 
w 
. 
X 
. . 0 
0 
A u 2Hd*X- 
-2(aa x +..)P . . o 
'H, 
t 
H 
1 
>< 
. . 0 
0 
Uy 
P 
. . 0 
0 
w' 
• 
. . 0 
0 
v' 
• 
. . 0 
0 
. 
X 
. . T>t 
Jet 
A u 2HB x X- 
.2(2M-..)P . . 0 
% 
and thence to 
