496 
[85 
85. 
NOTE ON A FAMILY OF CURVES OF THE FOURTH ORDER. 
[From the Cambridge and Dublin Mathematical Journal, vol. v. (1850), pp. 148—152.] 
The following theorem, in a slightly different and somewhat less general form, is 
demonstrated in Mr Hearn’s “Researches on Curves of the Second Order, &c.”, Loud. 1846: 
“ The locus of the pole of a line, u + v + w = 0, with respect to the conics passing 
through the angles of the triangle (u = 0, v = 0, w = 0), and touching a fixed line 
an. + ¡3v + 7w — 0, is the curve of the fourth order, 
V [au (v + w — w)} + \/ {/3v (w + u — v)} + V [yw (u + v — w)} = 0 ; ” 
the difference in fact being, that with Mr Hearn the indeterminate line u + v + w = 0 
is replaced by the line oo, so that the poles in question become the centres of the conics. 
Previous to discussing the curve of the fourth order, it will be convenient to 
notice a property of curves of the fourth order with three double points. Such curves 
contain eleven arbitrary constants: or if we consider the double points as given, then 
five arbitrary constants. From each double point may be drawn two tangents to the 
curve; any five of the points of contact of these tangents determine the curve, and 
consequently determine the sixth point of contact. The nature of this relation will be 
subsequently explained; at present it may be remarked that it is such that, if three 
of the points of contact (each one of such points of contact corresponding to a 
different double point) lie in a straight line, the remaining three points of contact also 
lie in a straight line. A curve of the fourth order having three given double points 
and besides such that the points of contact of the tangents from the double points 
lie three and three in two straight lines, contains therefore four arbitrary constants. 
Now it is easily seen that the curve in question has three double points, viz. the 
points given by the equations 
(u = 0, v — w = 0), (v= 0, w — u = 0), (w = 0, u — v = 0), 
points which may be geometrically defined as the projections from the angles of the 
triangle (u = 0, v = 0, w = 0) upon the opposite sides, of the point [u = v = w) which is 
the harmonic with respect to the triangle of the given line u + v + w = 0. Moreover,