TO — 1 points 
iodal Residue 
[n — 1) points ; 
single point ; 
m, n). The 
iltiple curves, 
L)tuple curve ; 
^ [to] 2 + M —1 
le construction 
mg line meets 
curve, these 
et the Nodal 
of points in 
ymbol for the 
-2, 
r. 
which gives 
II (to 3 ) = ^ [to] 3 — f [to] 2 + 3to — 5 + M (to — 5). 
The foregoing expressions for II might with propriety have been inserted in the Table. 
Annex No. 1.—Investigation of the formula for S (to 3 ) in the case of the unicursal 
curve (referred to, Art. 39). 
Consider the unicursal TO-thic curve the equations whereof are x : y : z:w = A : B: G: D, 
where A, B, G, D are rational and integral functions of a parameter 6; and let it be 
required to find the equation of a plane meeting the curve in such manner that 
three of the points of intersection are in lined. Taking for the equation of the plane 
%X + 7}y + + 0)W = 0, 
we find between (£, v], f, a>) an equation of a certain degree in (£, 7), £, «), which is 
the equation in plane-coordinates of the scroll S (to 3 ), the degree of the equation is 
therefore equal to the class of the scroll; but as the class of a scroll is equal to 
its order, the degree of the equation is equal to the order of the scroll, or say =8(m 3 ). 
Proceeding with the investigation, if 9 be determined by the equation 
+ 7)B + £C + (oD = 0, 
then the roots 9 1 , 0. 2 , ... 9 m of this equation belong to the points of intersection 
of the plane and curve; and the corresponding coordinates of these points are 
(A» 5i, G 1} A), &c. 
Suppose that the points 1, 2, 3 are in lined, and let X, ¡i, v, p be the coordinates 
of an arbitrary point, then the four points are in piano, that is, we have 
and if we form the equation 
X , 
P » 
v , 
p 
= 0; 
A 1} 
Bi, 
A, 
A 
A % , 
B 2> 
Ct, 
A 
A3, 
B 3 , 
G 3 , 
A 
X , 
p > 
v , 
p 
= 0, 
A, 
B x , 
A, 
A 
A a , 
B 2 , 
A, 
A 
A3, 
B3, 
A, 
A