256 
PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
Substituting in it for n x , v x the values which can be obtained from 
i (y v _ UW) = 0, and - (Uu - YW) = 0, 
WY, 
+ 
YW, 
— uU x 
UW, 
uv. 
2 
YW, 
u,w + 
UW, 
\ 
> 
1 
vu. 
UV 
V 
y„ T 
V TT 
- u 
- w 
-Y 
u 
u 
u 
II 
WY, 
+ 
YW, 
— wU, 
UW, 
UY, 
vw. 
uw,+ 
u,w 
- Y X V 
YU, 
u 
w 
. 
V 
WY, 
+ 
vw. 
UW, 
+ WU, 
uv,+ 
YU, 
2 
= wU 
YW, 
UW, 
+ WU, - 
-Y X V 
vu. 
u 
w 
V 
WV, vY x UV, 
YW, UW, + WU, — Y x v YU, 
u W Y 
W v U 
= YW, UW,. + WU, - Y x v YU, 
u W Y 
= 0 by (158). 
Next take 3 3 A /dxdy from (149). 
The first three determinants vanish by (158). 
The fifth and seventh determinants 
= Y, ^ (UW - Yv) + U, ~ (VW - \Ju) + w x ^ (uv - W 2 ) = 0. 
The eighth and ninth determinants 
= v 4 (UW - Vv) + U, 4 (VW - U u) + w,l(uv - W) = 0. 
