228 PROFESSOR M. J. M. HILL ON THE LOCUS OF SINGULAR POINTS 
(B.) If there be a conic node locus, then, besides equation (142), the following 
must be satisfied— 
+ 2W x ab + v x b 2 + 2Y x a + 2U x h -j- w x = 0 . . . . (161), 
Uyd 3 2W ydb -j- Vyb" -j" 2 Yyd -f" 2U yb -j- Wy = 0 . . . . (162), 
u.a 2 + 2W ,cib + vjr + 2 Y z a + 2U.6 w z — 0 . . . . (163). 
(C.) If there be a biplanar node locus, then the equations (126) are satisfied as well 
as the preceding. 
In this case these are 
2 (u x a + W x b + Y x ): 2 ( u y a + W yb + Y y ): 2 (u z a + W : b + Y ,): 2 u : 2W j 
= 2 (W x a + v x b + U,): 2 (W y a -f v y b + U y ) : 2 (W z a + v z b + U*): 2 W : 2vJ ^ ^ 
From these, the following may be deduced. 
Introducing a quantity \, such that 
u x a + W x b -fi Y x = Xu .(165), 
it follows by (164) that 
W x a -j- v x b 4 - IT* = \W.(166). 
From (165) and (166) 
u x a 2 4- 2 W x ab + v x b 2 4- aV* + b\J x = X(au 4" bW). 
Hence by (143) and (161) 
Y. r a 4- U x b 4 - w x = XV .(167). 
Similarly, quantities /x, v exist, such that 
and 
Uyd 4" W yb 4" Yy = 1-lU 1 
W yd 4" Vyb 4" Uy ~ fx\\ t 
Yyd 4- 11 yb 4- Wy = l-i \ J 
u z a 4- W M 4- V* = vu ~| 
W z a + vjb 4- U, = vW > 
V.a + U -j- w. = i\ 
(16S), 
(169). 
Consider now the equations (143), (144), (155), (165), (166), (167); multiply (143) 
by — v x , (144) by W. r , (165) by — v, (166) by W, and add. 
Therefore 
a (2WW* — uv x — vu x ) — Yv x 4- U W,. — vY x 4- WU* = A. (W 2 — uv). 
