Conies which pass through four given Points. 363 



infinity, is the required curve of foci. Taking U + XV=0 for 

 the equation of a conic of the series, the pair of tangents from I 

 is given by an equation of the form 



(X, l)*(»,y, *)« = 0, 



and- the pair of tangents from J by an equation of the like form 



and by eliminating X from these equations, we obtain the equa- 

 tion 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 conies there are two conies each of them 

 touching the line IJ, and that, considering the tangents drawn 

 to either of these conies, 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 conies shows that each of the points I, J is a double 

 point on the locus. 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), (AC.BD), (AD.BC) is a double point on 

 the curve : this establishes the existence of five double points. 

 The two conies of the series which touch the line IA are a single 

 conic taken twice, and the consideration of this conic shows that 

 the line I A 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 tan- 

 gent 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 num- 

 ber 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 

 (2 x 4 + 2 = )10; that is, I being a double point, the class of the 

 curve is =14; and assuming that the depression (6.5 — 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 conies 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 I 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 I to the curve ; that is, the 

 number of tangents from I to the curve is =2. 



Considering I, J as the circular points at infinity, and writing 

 A, B, C, D to denote the squared distances of a point P from the 

 four points A, B, C, D respectively, then, as remarked by Pro- 



