NEWSON: PROIECnVF. IRANSKORMATIONS. 



53 



formations througli B and along 1 respective!}'. When type II 



degenerates to type III, sin<^ and d become zero at the same time 



,1 / ,• smd> . . . ■ J , 



and we have a==hm. ar=u(i; a is so tar undetermmed but 



d 



will be shown later to have an important geometric meaning. 



15. The Invariant Conic K. Let T" be a transformation of 

 type III leaving the invariant the lineal element Al, Fig. 6, and 



transforming P to P^ and Pj to P,; 



suppose also that Q on Al is trans- 

 formed to Qj and Q, to Qo- In 

 the invariant pencil through A we 

 liave cotPjAl — cotPAl^cotPgAl — 

 cotP,Al=a. Along the line Al we 



1 ^ ^ I If 



have ^ -^= -—:^ -=:a 



AQ.3 AQ, AQ, AQ 



with the relation that a'=;/xa. 



Let K be the conic touching Al 



at A and passing through PjP^ and P^,. If Q be the point where 



the tangent to K at P cuts 1, then the tangents at P, and P^ will 



cut 1 at (.), and Oo respectiveh'. Taking the origin at A and the 



axis of X along Al, the equation of K may be written 



ax - — 2hxy + by - =2y . 



Let the coordinates of P,Pj, and P._, be (x',y'), (x,,y,) and 



(Xg,y^) respectively. The tangent to K at P is given by 



axx' — hxy' — hx'y ' byy':=y-^y'. 



This line cuts Al at O, so that 



Ag= ,'■■, ,. 



ax — hy 

 In like manner the intercepts on Al of the tangents at P, and P^ 

 are respectively 



and -; 



ax J - hy I ax„ — hy^ 



These intercepts must be AQ, and AQ., respectively; for we 

 have, taking the difference of their reciprocals, 



, aXa — hy, ax, — hy, ax, — hy, ax' — hy' 



« _ - - ^^ _ — ~ _ 



y 



y, 



H^-^)-K^-::h 



X X 



since - =cotP.,.A], — 

 v.. ' V 



cotP, Al and 



icotPAl. But we know 



