ME. W. SPOTTISWOODE ON THE CONTACT OF CONICS WITH SURFACES. 297 
-d, a, 
u * ’ 
P 
a', 
P 
u, 
V 
b,v 
=-B„ a, 
P 
a', 
P 
u, B^y = 
» b,v 
3H A a 
(3 
> a s', 
P V 
B.V 
7 
y' 
w 
a.v 
7 
7 
w 
7 
y w 
h 
V 
k 
h 
l' 
a B,y 
& 
y £ 
B,y ; 
or taking the first and last of these expressions, and substituting from (26), 
— {a 
u 1 
a! 
w, 
B,y+<* 
I 
CC 
w b* vy 
= a a! p 
B.V 
P 
P 
w' 
*,v /3 
p 
« BAV 
P P 2 
7 
y 1 
V 1 
b,v y 
1 
7 
w d,B,V 
y y' r 
y 
U 
b,v & 
&' 
k b.b,v 
l V s 
B t y; 
or substituting 
■ du for c^V, . . 
9 
3H 
cc ' 
u x 
a! 
m B* V } = 
-a a! p 
u 
p 
p 
w 1 
w /3 
P 
« b,b,v 
P P 2 
V 
7 
y 
V 1 
w y 
y 
w B,B r V 
y y' r 
w 
& 
V 
V 
k l 
y 
£ b.b,v 
i y 5 
k, 
or 
a 
a' 
u (up — 
3H?q)B,y + 3m^ v=o 
■ 
• (27) 
P 
P 
v (uq — 
3Hw')B, v + 3IM A v 
7 
y 
w (ur— 
3Hy' )d,V+3Hw3AV 
and if 
& 
S' 
k (us — 
3 HZ' )B,y + 3H^,B,V ; 
V: 
= (a, 
b, c, d, f, 
g, h, 1, m, n)(x, 
2/, 2 , *) 2 , • 
• • (28) 
the expression (27) is equivalent to 
a {a a! u (up— +h{. .}+g{. .}+!{. .}=0 . . (29) 
/3 /3' v (uq-oHw')x 
y y' w (ur — 3117/ )x 
S l 1 k (us — -3HZ' )x, 
which contains only the four coefficients a, h, g, 1 . In this expression the coefficient 
a = 7J5-, 
• 
2H u 
=B, 
u , 
3Hw'— 
•2H u (3, 
P, 
■y 
/3, 
V, 
uq— 3Hw' 
c, 
w, 
wr— 3HV 
7, 
y> 
w 
y> 
y 5 
U), 
ur — 3Hy' 
D, 
A, 
%s-3HZ' 
A 
a, 
y. 
ws — 3HZ'. 
2 E 
MDCCCLXX. 
