282 
ME. W. SPOTTISWOODE ON THE CONTACT OF STTEFACES. 
This being so, let 
a , 
P , 
7 5 
i , 
=T, 
C4 , 
P , 
7 i 
* , 
— T 
— x 11 
a , 
P , 
7 1 
i 5 
II 
u! , 
P'i 
T' 5 
y, 
cc' , 
P'i 
r / » 
V, 
a' , 
P’i 
7' 1 
y, 
u , 
V , 
w , 
« 5 
v , 
w , 
1c, 
v , 
Pm 
Om 
r n , 
P\2i 
0.121 
9*121 
S\21 
p221 
O 221 
9 225 
S 22 i 
a , 
P , 
7 » 
i , 
=0 2 , 
a , 
P 1 
7 i 
i , 
II 
1 
to 
p 
a , 
P , 
7 5 
l 1 
1 
=0. 
u , 
V , 
w, 
« , 
V , 
w , 
u , 
V , 
w, 
1c , 
i 
i 
125 
0m 
9*121 
S 125 
P-ni 
0-221 
9*221 
^ 2 2 , 
Pn, 
0m 
^1.5 
^115 
P 221 
0^21 
9*221 
^225 
Pm 
On, 
r U i 
^115 
i>i» 
0^21 
^*125 
^125 
; 
Also let 0 2 , O',, O' represent the expressions obtained by writing a', /3', y', o' for 
a, /3, y , h respectively in the first lines of 0 2 , 0„ O. Then it will be found, by the method 
of compound determinants, that 
-^■1115 
Y 
X 1115 
=20 
75 
+ T 
W , & , 
= 
7 5 
i , 
T, 
X,, 2 , 
Y 
x 112? 
w, k. 
^ 115 ^115 
w , 
1c 1 
• 
^ 115 
^115 
20, 
Avhence 
x 
^1115 
Y 
x in? 
Z„ >5 
= 
2T, 
i , 
T, 
= 2^{2T 1 0-T(0 1 +0')}; 
X, 12 , 
Y 
x 112? 
Z]]25 
• 
A, 
• 
X 
J v 1225 
Y 
x 122? 
y 
* J 1225 
2Q f +20 I , 
^115 
20, 
= 2£(r a Q-TQ' 1 ); 
both of which, when equated to zero, are comprised in the system 
T 2 T, ~T 2 ' ’ 
Similarly, taking the determinants 
x 
- /v 1125 
Y 
x 112? 
y 
/J 1125 
= 0 , 
X m , 
Y 
x im 
z m , 
X, 2 25 
Y 
x 122? 
y 
*0225 
X 122 5 
Y 
x 122? 
^1225 
-^2225 
Y 
x 222? 
y 
*02225 
X 222 5 
Y 
x 222? 
y 
* J 2225 
we should arrive at the system 
Oi_0' 1 + H 2 __H , 2 
T 2 Tj T^' 
similarly, 
X m , 
Y 
X 1115 
Z m , 
X,, 2 , 
Y 
X 1125 
y 
^112? 
x 
■^ x 222? 
Y 
x 222? 
7 
/J 222? 
( 70 ) 
