& 



INVOLUTIVE 1-1 QUADRIC TRANSFORMATION IN A PLANE. 261 



§ 11. Numerous examples of either transformation are to be found in the elementary 

 geometry. One example of each is given. 



If the base A A' of a triangle A A' C is kept fixed, the orthocentre P of the triangle 

 is such that C is the orthocentre of the triangle A A' P. Hence to C corresponds P and 

 to P the point C. Moreover, if C moves in a straight line, the locus of P is in general 

 a conic. The transformation is therefore one of the kind in question. 



Take A A' as the cc-axis, so that A and A' are the points (a, 0), ( — a, 0). Let C 

 be the point (£, rj) ; then P has for co-ordinates 



Hence the two pairs of co-ordinates are connected by the relation 



x— £=0; yt]+x£—a 2 = 0. 



Hence the straight line C P passes through the point at infinity in a direction 

 perpendicular to A A', and P is on the polar of C with respect to the circle whose 

 diameter is A A'. Hence the transformation is a Hirst transformation. 



The analysis also leads to the known proposition that the three lines found by 

 taking the polar of each vertex of a triangle with respect to the circle which has for 

 diameter the opposite side are concurrent in the orthocentre of the triangle. 



§ 12. A Beltrami transformation is furnished by the following theorem of Professor 

 Chrystal's, and its generalisation, viz. : " A circle meets the side B C of a triangle 

 A B C in D and D', C A in E and E', and A B in F and F'. If A D, B E, C F be 

 concurrent, then A D', B E', C F' are also concurrent. 



Tins is included in the following : P and P' are two points taken in the side A of 

 the triangle A B, and Q Q' on the side B such that P.O F = p, Q.O Q' = <r, where 

 p and a are constants. A Q and B P meet in S ; A Q' and B P' meet in S'. If A B is 

 met by S in R and by S' in R', it follows that the six points P P' Q Q' PR' lie on 

 a conic. Moreover, to S corresponds a unique point S' and inversely to S' corresponds 

 S, so that the transformation from S to S' is involutive. Also if S move on a straight 

 line S' in general traces out a conic. The transformation ought therefore to be either 

 a Hirst transformation or a Beltrami transformation. 



To obtain the transformation, let OA and OB be taken as axes, and let A = a, 

 B = b, P = a, Q = /3 (so that a and /3 vary). 



Then A Q and B P have for equations 



a & I (i.) 



Hence, if S be the point (x, y), 



a = 





bx n ay /•• x 



b—y' a — x 



VOL. XL. PART II. (NO. 13). 2 Q 



