AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
231 
Hence in the case of biplanar and uniplanar node loci, these six determinants are 
by (170)—(17 5) 
= ( pa 3 + 2Ra& + qb 2 + 2Qa + 2P b + r) ^ (uv — W 3 ). 
(ii.) 
u x 
w. 
V. 
w. 
v* 
u. 
V, 
u. 
W 3 ) . 
u x W, 
<1 
*4 
Wf ctu x -f- + V* 
w, 
u. 
= 
w 
* V x «w, + + U, 
v, u. 
w x 
y* u, + b\j x -f- w x 
u x 
w. 
u 
= X 
w. 
v* 
w 
V, 
u. 
y 
u x 
W, 
w, 
au x + bW x + V, aW, + bv x + XL 
u 
W 
au -f- bW V 
(178) 
u x 
w. 
w. 
w 
w 
0 
= X 2 ( - u, W 3 + 2ttWW.« - u%) 
= X 3 ( — u x uv + 2wWW, — u 2 v x ) 
= - \ 2 u ~ (uv - W 3 ). 
w. 
V 
v y 
% 
w 
VV y 
V 
v 
Ujt 
w, 
V, 
w. 
u. 
+ 
w* 
V g 
u 2 
4- 
w. 
TT, 
y* 
U, 
«y 2 
v« 
v< 
iL 
V, 
u. 
“V 
u x 
w. 
v = 
u x 
w* 
v. 
w. 
V, 
w. 
u. 
+ 
w. 
% 
U, 
4- 
w. 
a 
V, 
u. 
Wx 
v 2 
u 2 
U, 
Wy 
- fyvu ~ (uv - W 3 ) - 2i/\w ^ (w - W 3 ) - 2\/xu ~ (uv - W 3 ) . (179). 
