348] 
ON THE THEORY OE INVOLUTION. 
301 
which is Prof. Boole’s ancient theorem referred to in my Fifth Memoir on Quantics ( J ), 
but which is now first exhibited in connexion with the general theory to which it 
belongs. 
16. It may be noticed that the condition for a twofold critic centre, or (what is 
the same thing) a twofold factor of the Jacobian {which condition is of the degree 
2 (2n — 3) in regard to the coefficients of TJ or F} is RQ — 0; and that we in fact have 
2(2n-3) = n+3(n-2). 
This remark is due to Dr Salmon. 
Article, Nos. 17 to 42, relating to two Ternary Quantics. 
17. Suppose now that Z7= (a, ..{$#, y, z) n , V=(a',.R[x } y, z) n are two ternary 
quantics of the same order n, and write W = U + kV — (a + ha,. Q$x, y, z) n , so that 
W=U + kV= 0 
is the equation of a curve in involution with the curves U = 0, F = 0. But for 
greater distinctness it is in general proper to retain U+fcV in place of W. 
18. In order that the curve U+kV = 0 may have a node, we must have 
simultaneously 
S x (U+kV) = 0, 
8 y (U + kV) = 0, 
8 z (U+kV) =0, 
and eliminating (x, y, z) from these equations we have 
Disc 1 . (U+kV) = 0, 
which is an equation of the degree 3 (n — l) 2 as regards k, and of the same order as 
regards the coefficients of U, V conjointly. 
19. To each of the S(n— l) 2 values of k there corresponds a point satisfying 
the conditions in question, and which is therefore a node of the corresponding nodal 
curve 
U + kV = 0 ; 
the points in question are the critic centres of the involution. 
20. The critic centres may be differently obtained as follows; viz. if from the 
three equations we eliminate k, we find 
KU, ZyU, S Z U 
B X V, 8yV, 
= 0, 
1 Pliil. Trans, vol. cxlviii. (1858), pp. 415—427, [156].