AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 209 
k f] <8f r-+h.,] (hr- + K, a m 1 
+ 2 h, (] ( 8 ,) (Si) + 2 B, f] (SO (Sf) + 2 [(,,] (Sf) ( 8 ,) = 0 . . ( 120 ). 
and then making the roots of the resulting quadratic in Sa/S/3 equal. 
For making the substitution, the result is 
(§a ) 2 {[£ £] Xf + [77, Tj] pp + [£, Q v { 2 + 2 [77, Q PiJq + 2 [£, £] jqXj + 2 [£ 77] X^} 
+ 2 (Sa) (S/3) 
[£ £] X i X e + bi> y] MiM 2 + [£> Q v \ v 2 
+ bi> £] (/lmj + w\) + [£. f\ biK + + [£ yl (^2 + \*/*i) 
+ (<W {[£ £] V + bi> v] i x i +F£> £] v i + 2 [ 17 , C\ p^ + 2 [£. £] ^. 2 x, + 2 [£ 77 ] x 2 p*} = °- 
Now putting 
L i = [£ f] x i + [£ mi + [£ £] *q, 
M i = b> £] x i + b)> y\ \ x 1 + bh Q v \-. 
N i = [£, £] + [£> >?] /l + [£> C] v x , 
and similar expressions for L 2 , M 2 , N 2 , the condition for equal roots can be written 
X l Li + pjMj + *qNi X]L 2 -b p^ 3 iqN 2 j 
I X 2 L x ~b p 2 Mi + voN^ X,L, + p 2 M 2 + e 2 N 2 — 0. 
It remains to show that this will be satisfied if (119) be satisfied. 
Now 
K £] 
a 
Rfl 
Ml ^2 “ 
P2 z 'l 
Xi 
F-i 
0 
l&y] 
h> > 7 ] 
It U 
EjX 2 — 
a 2 Xj 
X 
Xo 
/L 
^2 
0 
BC] 
b, C] 
[t £] 
^•l / x 2 
0 
0 
1 
0 
t x l v 2 ~ l x 2 v \ 
Z^q^-p 
X]p 2 ^ 2 pl 
0 
0 
0 
0 
I 
Mi 
Ni 
0 
Mi 
0 
L 2 
m 3 
n 2 
0 
X 
X 2 
p 2 
^2 
0 1 
[t« 
K,^] 
k. a 
^•j / A 2 
X,pi 
0 
0 
1 
0 
Pl^o — Po^ 
^ 1 X 2 1 ^ 2^1 
Xp^Xo ^2/^1 
0 
0 
0 
0 
1 
X 1 L 1 -fi pjlVIj jqNj X^L, -b p^l\f 2 -b iqN 2 N^ 0 
XoLj + p 2 Mi + zqN i X,L 3 ~b p 2 ]\I 2 + zqNg N 3 0 
[£> £] ^#2 — X 2^1 
^■l / x 2 ® 
X l L 2 + p x M 2 + *qN 2 
X L, fi- p 0 M 2 + t'gNg I 
N n 
(X 1 P 2 “ x 2Mi)' 
N 2 
0 
X1L1 + /.q Mj -b ^N x 
X 2 L x ~b p 2 X 1 y -b zqNj 
2 E 
MDCCCXCII.—A. 
