AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 193 
Put therein <f> ( u , v) — Df/Du, xjj (u, v) = Df/Dv. 
Then the condition of Preliminary Theorem C, which also holds, is 
Dy Wf _ / p y y 
Dm 2 Di? 2 Dv) 
The condition of Preliminary Theorem D is equivalent to 
py/ py 
D id \ Du Dv 
Dy d y py py /pyy 
DADw Dm Dv Dm 2 D« Dv 2 \Dm 2 / _ 
pyy 
D v*J 
py /py\ 3 _ o py py py , py / py 
Dv? Dv \Dv 2 / ~ 2 Du Dv 2 Dv 2 Dm Du + Dv> \DtiP-, 
py y 
DwDvJ 
Dividing out by ( D~f[Du Du) 3 the last equation becomes, by the preceding, 
Dy/pyy o d 3 / d 2 / /d 2 A 2 py / Dyyp 2 / py / Dy \ 3 
Did \Dv 2 / _ 3 Dm 2 Do Du Dv [d?J + 3 pk pD \Dm Dv) Dv 2 ~ Dv 3 \DmD«/ 
Now in the former part of this article, a and /3 correspond to u and v. 
Hence the condition that the equations D//Da = 0, Djf/D6 = 0 may have three 
coinciding systems of common roots is 
[a, a, a] [/ 3 , /3] 3 — 3 [a, a, /3] [/3, [a, /3] + 3 [a, /3, fi ] [/3, /3][a, /3J — [/3, /3, £] [a, /3] 3 = 0. 
This fact must be taken account of in forming the discriminant, and the whole of 
the work must be modified in accordance with it. But this case will not be further 
discussed. 
(C.) It has now been shown how to determine the values of da/dx, db/dx, when 
x = £, y = rj, z — £, the coordinates of a point on the uniplanar node locus. 
Now, when x — g, y = rj, z == £, 
rt 7 \ df (x,y, z, a v l J 
/ (a y, 2, oq, A), -^- 
both vanish. 
dy (x, y, a, a 1 ,b l ) d /df(x, y, z, a v 6J 
Hence when x = A y = rj, z = 
MDCCCXCII.—A. 
0£ 
9a: 
_ 0 / D/ Q, y, z, a v p) 
dx ^ Dx 
r- —| |—■ -. dll. . r 7 0k 
= [x, x] -p [x, Gq] 0^ + \_X, A] 0-p . 
