348] 
ON THE THEORY OF INVOLUTION. 
303 
25. First, if the curves ¿7=0, V— 0 touch each other, then, (x, y, z) being the 
coordinates of the point of contact, we have ¿7=0, V= 0, 
8 X (U + k x Y) — 0, 
8 y (¿7 + V) = 0, 
8 z (U + k 1 V) = 0, 
where k^ denotes the value given by the equations 
8 X U : 8 y U : 8 Z U = 8 X V : 8 y V : 8 z V=k, : -1 
belonging to the point of contact. It at once follows that every diacritic curve passes 
through the point in question. But it is somewhat more difficult to show that the 
diacritic curves touch at this point. 
26. I represent for shortness the first and second differential coefficients of ¿7 by 
(L, M, N), (a, b, c, f g, h), and similarly those of V by (U, M', N'), (a', b', c, f, g\ h'), 
these values all belonging to the point of contact: we have therefore 
L + kjl = 0, M + k-M = 0, N + kJF = 0. 
The equation of the diacritic curve is 
« > /3 > 7 
L, M, N 
L', M', N' 
= 0; 
to find the tangent we must operate on the left-hand side with X8 x + Y8 y + Z8 z , where 
X, Y, Z are current coordinates. Calling the foregoing symbol D, this gives 
a , 
P 
7 
+ 
a , 
P , 
7 
= 0; 
L , 
M 
N 
DL, 
DM, 
DX 
1)IJ, 
DM', 
DX' 
u, 
M’, 
X' 
thing, 
a , 
¡3 
7 
- 
a , 
/3 , 
7 
= 0; 
L , 
M 
X 
u, 
M', 
X' 
DU, 
DM', 
DX 
DL, 
DM, 
DX 
le first 
actor k 
determinant for L, M, X their values — kU, — kM', — kX', 
from the second to the third line, we obtain 
* 
/3 , 
7 
- 
a. , 
/3 , 
7 
= °, 1 
u , 
M , 
X' 
u, 
M, 
X' 
kJ)U, 
k x DM, 
kjDX' 
DL, 
DM, 
DX