( 765 ) 
", SL es Yee 
Putting £= * and taking into account the condition (3) the first 
ls 
equation (¢), in which the w represent the coordinates of the given 
point P, proves to be of order 2m 4 n +1) in #, showing that 71 is 
of order 2(m-—+ n +1). 
The lines, the corresponding conics of E of which pass through a 
given point P, form a complea IT of order 2m 4 n +1). 
4. Let us now consider a point P and a line p passing through 
it. Then the conics of / passing through P and lying in planes 
through p correspond to the lines of a congruence a contained in 77. 
We immediately see that the congruence a is of the first order. 
For, in determining the number of lines of this congruence passing 
through any other point Q, it is evident that the plane (u) is com- 
pletely determined by the conditions that it must contain line p and 
point Q. In this plane the lines to which correspond conics passing 
through P form a pencil, only one ray of which passes through Q. 
So we see too, that the line p is singular for the congruence. 
Let 2, —7, 0" be the equations of the lime pand rz =d —0 
We 
the coordinates of the point P?. Let us put 4—=— and deduce the 
u 
1 
equations of a conic of / passing through P and lying in a plane 
through p; let 
7; art + ke, = 0, 
be the resulting equations. Here 7=0O, derived from the first equation 
(e) of art. 1, is of order 2m 4-n) +1 in £ and linear in (a, a,, a;, a). 
Let us suppose £,==0 and introduce moreover the supposition 
a, = 0 into the equation f= 0. The centre of the pencil formed by 
the lines of the plane 
ets ke, == Ot oe LNE eh Sy (4) 
where £20, to which correspond conies passing through P, is 
determined by equation (4) and by 
b, x, = b, ==, GA bres 
in connection with 
Ls b= 0} PE Ok 
From this we conclude that the second singular line of a is a 
twisted curve of order 4(0m--n-+1)-+ 1 cutting the line p a number 
of 40m + 7 + 1) times (in the neighbourhood of 7, = 0 this curve is 
determined by continuity). 
