232 
PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
For multiplying the first column in each of these determinants by a, the second by 
b, and adding to the last they become 
u y 
W, 
[XU 
Uy 
yU 
u. 
w 2 
vu 
w. 
v x 
xw 
+ 
w 2 
v z 
vW 
+ 
w. 
v x 
XW 
V, 
v z 
vY 
v. 
v x 
XV 
V, 
Vy 
^xV 
u. 
w s 
vu 
u x 
w. 
Xu 
u x 
w x 
Xu 
w 
vv y 
v y 
/xW 
+ 
w„ 
Vy 
/xW 
+ 
w, 
v z 
vW 
v x 
u. 
XV 
V, 
Vz 
vY 
u. 
y\ 
The coefficient of X is 
Uy 
W, 
0 
u~ 
Wz 
u 
u. 
w : 
w 
Wz 
Vz 
w 
+ 
w ; 
Vz 
W 
+ 
W , 
0 
Vz 
u. 
V 
u, 
0 
V, 
u. 
V 
The coefficient of /x is 
w* 
w. 
u 
v x 
W. E 
w 
u. 
w ; 
0 
W. 
w 
+ 
w , 
V; 
0 
+ 
w. 
P.r 
w 
v* 
Vz 
0 
V, 
u. 
V 
V, 
H, 
V 
The coefficient of v is 
Uy 
w y 
Uy 
w„ 
U 
W* 
W. t 
0 
W, 
0 
+ 
' y 
w 
+ 
W, 
Vy 
w 
V, 
u. 
V 
V., 
u. 
0 
U y 
V 
The coefficient of /x can be obtained from that of X by changing 2 into x and y 
into z. 
The coefficient of v can be obtained from that of X by changing y into x and z 
into y. 
Hence it is sufficient to calculate the coefficient of X. 
The coefficient of X, viz. :— 
