54 KANSAS UNIVERSITY QUARTERLY. 



that a'=fjia; and if we put )u,=a, it follows that the transformation 

 . T" which transforms P into Pj and Pj into Pg also transforms Q 

 into Q, and Q, into Q^; Q, Q,, and Q., being the intersections 

 with Al of the tangents to K at P, P^, and P„. It follows immedi- 

 ately that the conic K is transformed into itself, for it is transformed 

 into some conic K' which must touch Al at A, since Al is an invari- 

 ant element; since PQ is transformed into P^ Qj andPjQ^ intoPgQ^, 

 it follows that K' must touch P,Q, at P, and PgQg at Po and hence 

 K' is the same conic as K. 



Theorem J. The transformaiion T" whicli leaves invariant tJte 

 lineal element Al and transforms P into P^ and F ^ into Po also leaves 

 invariant the eonie determined by Al, P, P, and Pg. 



16. Invariant Relations. From the relation cotP, Al=;COtPAl-(-u 

 we see that 



^i_=_-JL+„. (,4) 



yi y 



where (x^,y,) and (x,y) are the coordinates of P^ and P in rectan- 

 gular coordinates. From this relation and the equation of the 

 invariant conic through P and P, we can deduce another important 

 relation between the coordinates of P and P,. Solving the equa- 

 tion ax- — 2hxy-fby--=2y as a quadratic in x we have 



The same relation holds for Xj and y,, viz: 



= - — ha 1 h2_ab + ^ + ,, «-• 



y . 



Substituting for the radical its value, this reduces to 



I I /ax — hv\ a „ I , x a ^ , , , 



.. — L- a I ~ IH a-— h aa 1 a-— ha. (15) 



yjy \y/2 y y'2 



The two relations — !-= j-a and — = 1- aa 1 a^ — ha 



yi y yj y y 2 



