INVOLUTIVE 1-1 QUADRIC TRANSFORMATION IN A PLANE. 257 



§ 6. The analysis therefore leads to the conclusion that when the transformation is 



involutive the bilinear equations may be reduced to one or other of the types I. and II. 



If we consider (£, >?) as a fixed point, the equation I. is simply its polar with respect 



to the conic 



A 1 x 2 + 2B 1 xy + B$ 2 + 2C,x+ 2C,?y + C 3 = ; 



whereas the equation 



Cl ( x -£) + C 2 (y - V ) + B x (& - xn) = 



C 2 (V 



is simply the equation to the straight line joining (£, n) to the fixed point . 



V bj £>! 



Hence the theorem : — 



The most general transformation of the nature of a quadric inversion, in which to a 

 straight line corresponds a conic, may be obtained as a point transformation in ivhich :— 



First. — To a point corresponds the intersection of its polars with respect to two 

 fixed conies ; or 



Second. — To a point corresponds the intersection of its polar with respect to a 

 fixed conic with the straight line joining the point to a fixed point. 



The case in which there are two equations of the form II. simply corresponds to the 

 identical transformation. Naturally there are various degenerate cases, for many of 

 which Czuber's paper may be profitably consulted. The ordinary inversion is a 

 particular case of the second transformation in which the conic is a circle, while the 

 fixed point is its centre. 



Both transformations have already been discussed geometrically, the first by 

 Beltrami in 1863, in his well-known memoir, " Intorno alle coniche di nove Punti" 

 {Mem. della Acad, di Bologna, Tomo II.) ; the second by Hirst (" Quadric Inversion of 

 Plane Curves," Proc. R. S. L., 1865). 



Hirst never mentions the Beltrami transformations, although Beltrami had already 

 shown how to obtain certain Hirst transformations, such as the ordinary inversion. 

 Czuber (I.e.), in his analytical discussion, has omitted the Hirst transformations entirely. 



§ 7. The two transformations present several points of contrast, and that of Beltrami 

 would appear to be the more symmetrical. 



In the Beltrami transformation let the two conies be represented by 



ax 2 + if +cz 2 = 

 lx 2 + my 2 + nz 2 = Q, 



as referred to their common self-conjugate triangle X YZ. 

 To any point (£, *j, ^) corresponds the point given by 



ax£ -f bytj +cz£ =0 

 lx£ + myr\ + nz^ — , 

 therefore 



x :y :z = (hu — cm)t]g : (cl — an)££ : (am — &/)£"»; 

 and similarly 



£:iy.^—(bn — cm)yz: etc. : etc. 



There are four self-corresponding points, the points A, B, C, D, in which the two 



