2()0 Mil CHARLES TWEEDIE ON THE 



Let (£, ij, £) be the co-ordinates of P. The equation to the tangent at P is 



'(i'/,f+^+B, y + CO-l-.y(etc.) + <etc.) = 0. 



Hence, if this line pass through X, 



hut 2£+B,+C£=0 



.-. 2Af+2£ =0 



Hence & = —1, and the conic has for equation 



x- + ayz + x(Ey + Cz) = , 



while there is no distinction between the points P and Q. The theorem is therefore 

 established. Various sub-cases arise according to the relative positions of the four 

 common points. 



The tangents at Y and Z to the second conic are given by 



az+Bx=0 (1) 



ay + Cx = Q (2), 



and the tangent at P to the first conic is 



2zg-ay£-az 1] = (3). 

 These tangent lines are concurrent, provided 



which is the case. 



Hence the theorem : — 



" If two conies cut in Y, Z, P, Q, and if the pole of Y Z with respect to the first 

 conic coincides with the pole of P Q with respect to the second conic, then the pole of 

 Y Z with respect to the second conic coincides with the pole of P Q with respect to the 

 first conic." 



Naturally both statements admit of reciprocation. In a sense, they are particular 

 cases of the theorem that the eight tangents to two conies at their common points in 

 general envelop a curve of the second class. 



£ 10. It may be noted that one of the three canonical forms of the 1-1 quadric 

 transformation, as given in Miss Scott's Modern Analytical Geometry, 



_ 1 . 1 . 1 

 i 'i i 

 is a Beltrami transformation and not really a Hirst transformation. 

 It corresponds to 



so that (;/■, y, z) is the point of intersection of the polars of (£, 77, £) with respect to tin' 



two degenerate conies 



x 2 -y 2 =0; ^ 2 -^ = 0. 



The other two are Hirst transformations. 



