ON A CASE OF THE INVOLUTION OF CUBIC CUBVES. 
[From the Transactions of the Cambridge Philosophical Society, vol. xi. Part I, 
(1866), pp. 39—80.—Read 22 February, 1864.] 
The present memoir relates to the involution 
xyz + k (x + y+ z) 2 (Xx + py+ vz) = 0, 
viz. treating x, y, z as coordinates, and k as a variable parameter, this equation 
represents the series of cubic curves passing through the intersections of the two cubics 
xyz = 0, (x + y + z) 2 (Xx + py + vz) = 0 ; 
or, what is the same thing, the line x + y + z = 0 meets any cubic of the series in 
three points the tangents at which are x = 0, y — 0, z = 0, and these tangents again 
meet the cubic in three points lying on the line Xx + yy + vz — 0; so that in the 
language which I have used elsewhere, the lines x+y+z= 0, Xx + py + vz — 0 are in 
regard to the cubic a primary and a satellite line respectively. The investigation (which 
is a development of two short papers already published in the Philosophical Magazine)( x ) 
was undertaken in order to applying it to the explanation and discussion of Pliicker’s 
Classification of Curves of the Third Order; but such application will properly be made 
in a separate memoir, On the Classification of Cubic Curves, and it has also appeared 
to me convenient to give therein the discussion of the geometrical forms of certain 
loci which present themselves in the present memoir. 
I remark that the involution intended to be here considered is a case of the more 
general one U+kV= 0, where TJ = 0, F = 0 are any two cubic curves whatever. It appears 
from my memoir On the Theory of Involution, [348], that the equation, Disc 1 . (U+kV) = 0, 
which determines the critic values of h is in the general case of the order 12; the 
1 On the Cubic Centres of a Line with respect to Three Lines and a Line.—Phil. Mag. t. xx. 
pp. 418—428 (1860), [257]. Ditto, Second Paper, t. xxii. pp. 488—436 (1861), [315]. 
C. V. 
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