2 
ON THE LOCUS OF THE FOCI OF THE 
[417 
the locus of the intersections of the tangents from I and the tangents from J to any 
conic of the series; which curve, if I, J are the circular points at infinity, is the 
required curve of foci. Taking I7 + XF=0 for the equation of a conic of the series, 
the pair of tangents from I is given by an equation of the form 
O, l) 2 (x, y, zf = 0, 
and the pair of tangents from J by an equation of the like form 
(X, 1 ) 2 0, y, ¿0 2 = 0; 
and by eliminating X from these equations, we obtain the equation of the required 
curve. This in the first instance presents itself as an equation of the eighth order; 
but it is to be observed that in the series of conics there are two conics each of them 
touching the line IJ, and that, considering the tangents drawn to either of these 
conics, the line IJ presents itself as part of the locus; that is, the line IJ twice 
repeated is part of the locus; and the residual curve is thus of the order 8 — 2, = 6; 
that is, the required curve is of the order 6. The consideration of the same two 
conics shows that each of the points] I, J is a double point on the curve. Moreover, 
by taking for the conic any one of the line-pairs through the four points, it appears 
that each of the points (AB.CD), (AG. BD), (AD.BC) is a double point on the curve: 
this establishes the existence of five double points. The two conics of the series which 
touch the line IA are a single conic taken twice, and the consideration of this conic 
shows that the line IA is a double tangent to the curve; similarly each of the 
eight lines I (A, B, C, D) and J (A, B, C, D) is a double tangent to the curve. 
Instead of seeking to establish directly the existence of the remaining three double 
points, the easier course is to show that, besides the four double tangents from I, the 
number of tangents from I to the curve is = 2; for, this being so, the total number 
of tangents from I to the curve will be (2x4 + 2=) 10; that is, / being a double 
point, the class of the curve is = 14 ; and assuming that the depression (6 x 5 —14 =) 16 
in the class of the curve is caused by double points, the number of double points 
will be =8. But observing that in the series of conics there is one conic which 
passes through J, so that the tangents from J to this conic are the tangent at J 
twice repeated, then it is easy to see that the tangents from / to this conic, at the 
points where they meet the tangent at J, touch the required curve, and that these 
two tangents are in fact (besides the double tangents) the only tangents from / to 
the curve; that is, the number of tangents from I to the curve is = 2. 
Considering /, J as the circular points at infinity, and writing A, B, G, D to 
denote the squared distances of a point P from the four points A, B, G, D respectively, 
then, as remarked by Professor Sylvester, the equation 
xVM + /u ^B + v VC + 7rVD = 0 
(where X, ¡jl, v, nr are constants) is in general a curve of the order 8 ; but the ratios 
X : fji : v : nr may be so determined that the order of the curve in question shall be