296 
ON THE THEORY OF INVOLUTION. 
[348 
it being understood that the discriminant of the function U + JcV is taken in regard 
to the coordinates. And this being so, we may write Disc*. Disc 4 . (U + kV) to denote 
the discriminant in regard to k of the function Disc*. (TJ + kV). The quantity in 
question (viz. Disc*. Disc*. (U + kV), or say for shortness □) is a function of the 
coefficients of TJ, V, homogeneous as regards each set of coefficients separately, and it 
breaks up into factors in the form 
□ = RQ S P 2 , 
where R = 0 is the condition in order that the ranges U=0, V= 0 may have a point 
in common; or what is the same thing, in order that there shall be a range 
U + kV=0 having a twofold point at a common point of the ranges TJ = 0, V=0, 
(R is in fact the resultant of the qualities TJ, V): Q = 0 is the condition in order 
that there may be a range TJ + kV= 0 having a threefold point: and P=0 is the 
condition in order that there may be a range U+kV=0 having a pair of twofold 
points. 
And similarly, when U = 0, V—0 are curves, then we have the like equation 
□ = RQ 3 P 2 , 
where R = 0 is the condition in order that the curves U=0, V= 0 may have a point 
of twofold intersection, that is, that the two curves may touch each other, (R is the 
Tactinvariant of the quantics TJ, V); or what is the same thing, it is the condition 
in order that there may be a curve U+kV=0 having a node at a point of twofold 
intersection of the curves TJ = 0, V — 0; moreover Q = 0 is the condition in order that 
there may be a curve U+kV=0 having a cusp: and P = 0 is the condition in order 
that there may be a curve TJ + kV = 0 having a pair of nodes. 
The establishment and illustration of the foregoing theorems form the chief object 
of the present memoir. 
Article Nos. 1 to 16, relating to two Binary Quantics. 
1. Let TJ = (a, ...\x, y) 2 , V=(a',...\x, y) 2 , be two binary quantics of the same 
order n; and write W = TJ+kV —(a + ka, ..J$x, y) 2 , so that W=TJ+kV—0 is the 
equation of a range in involution with the ranges TJ = 0, V = 0. But for greater 
distinctness it is in general convenient to retain TJ + kV instead of replacing it by 
the single letter W. 
2. In order that the range TJ+kV= 0 may have a twofold point we must have 
simultaneously 
8 x (U+kV) = 0, 
8 y (U + kV)= 0, 
and eliminating (x, y) from these equations we find