230 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
(D.) If there be a uniplanar node locus, then in addition to the results obtained in 
(B) and (C), it follows by (48) that 
Vj r:r o?' -fi . . . ; Vj :ri jOj~ ~b ... i U, -Ci '~b . • • ’ 2 ( 7 i x ci -J - AV r Z) -b V.?-) • 2 (W x a -|- v x b -j - PR) 
— U T yCl>~ -}"•••• UyyQ/ d~ • • • • UyM~ -j~ • • • ' 2 (XtyCl -f- W yb fi - V y) t 2 (W yCl -j- Vyb -(- By) 
= u xz a 2 + . . . : u yz a 2 + . . . : u zz a 2 + . . . : 2 (u z a + W z b -b Y z ) : 2 (W.a + v 2 b + U z ) 
= 2 (u x a + AY x b + V. r ) : 2 ( u y a 4- W v b + V y ) : 2 (u.a + W z b -f Y z ) : 2 u : 2W 
= 2 (AY^n -f- vjo -f- U*) : 2 (AY y <x -j- Vyb ~b U y) : 2 (AY z n -b v z b -b U z ) ‘ 2AY : 2v (176). 
(E.) (i.) The following equations will be useful in the case of biplanar and 
uniplanar node loci:— 
p 
R 
Q 
V 
R 
Q 
«* 
w. 
Y* 
w. 
V x 
u. 
+ 
AY 
V 
u 
+ 
R 
q 
P 
V 
U 
w 
V, 
u. 
w x 
Y 
u 
w 
u 
AY 
V 
u x 
AY., 
Y. r 
u 
AY 
Y 
R 
q 
P 
+ 
AY 
V 
U 
+ 
w. 
v x 
u. 
Y, 
u. 
w x 
Q 
P 
V 
Q 
P 
r 
= (pa 2 -b 2B ab + qb° + 2Qa + 2P6 + r) £ (uv - AY 2 ). . . (177). 
For the first and second determinants 
= V l — U s ) + E l (UV - W„.) + Q | (WU - Yv). 
The third and fourth determinants 
= R a ^(UV - W«) + (uw - V*) + P^(WV - u«). 
The fifth and sixth determinants 
