262 Ml! CHARLES TWEEDIE ON QTJADRIC TRANSFORMATION. 



Similarly if S' be the point (£, rj), 



■P~r/ ■ P = * — ( 11L ) 



' at an v 



b — y bi; a—x an 



i.e., 1^-pQ>- y Xb-i) = 0]a*to-a(a-i)(a-v)=Q. (v.) 



Hence (x, y) is the point of intersection of the polars of (£, n) with respect to the 

 two degenerate conies : — 



BV _ p ( & _ y y = o ; a y - <r(a - xf = 0. (vi.) 



These conies determine a quadrilateral XYZW, which is such that the intercepts cut 

 off hy its sides on A and B are bisected at 0, while the two pairs of lines pass 

 through A and B respectively. For these conies may be substituted any two conies 

 through X Y Z W. 



Two of the principal points are A and B. The third principal point is not 0, but 

 the intersection C of X Z and Y W which are given by 



; cV(/o<r - a"-b 2 ) - ifp{pcr - a 2 b 2 ) - 2a po -<> - b 2 )x + 2bpa(p -a 2 )y + p(r (a?<r - b 2 p) = 0. 



(ABC is the self-conjugate triangle of the system of conies.) 

 The lines through the origin parallel to these are given by 



xV- y 2 p=0. 



They are therefore parallel to the sides of the parallelogram K L M N, where 

 KLMN are the points in which OA and OB are cut by the sides of XYZW. 

 Hence the third line-pair X Z and Y W are such that each makes on the axes 

 intercepts the square of whose ratio is p : cr. When p = <r, i.e., when the circles of 

 inversion coincide, the parallelogram is a rectangle, and each line of the third line-pair 

 makes equal intercepts on the axes. 



In the transformation as a Beltrami transformation the point has no important 

 role. In the general Beltrami transformation the lines through B are paired in involu- 

 tion, and they therefore determine a point-range in involution on any straight line 

 such as A. Similarly, the lines through A determine an involution on B, and we 

 have here the particular case in which the centres of involution of the point-ranges 

 coincide in the point common to the two ranges. If A and B are real, and if L N pass 

 through B, the locus of its middle point is a conic passing through A, and similarly 

 for as the middle point of K M through A. Hence when A and B are real there 

 may exist real points in finite number possessing the property in question. 



The transformation suggests the apparent generalisation : — A and B are fixed points 

 in the plane of two involutive point-ranges P P' and Q Q', where P P', Q Q' denote 

 corresponding points of the respective involutions. The lines joining A and B to 

 PP' and QQ' determine a quadrilateral STS'T' in which S corresponds to S' (or T to 

 T') in a 1-1 involutive quadric transformation. 



