( 198 ) 



To the points of intersection of P and A' belong the points of 

 contact of the six tangents from lUo k\ U' A^ is one of the remaining 

 tliree points of intersection, then .1., and A,, are harmonically separated 

 by A^ and P, that is P lies on the polar conic of A^ ; from this 

 follows however that A^ lies on the polar line of P. .So the curve 

 h'' })asses through the points of intersection of Z' with both the polar 

 conic jr and the polar line y/ of P. Its equation is therefore of 

 the form fio%-f «,/«"., ^'%/^>.r=0. If point A' belongs to the harmonic 

 curve of point Y, it is evident that Y lies on the liarmonic curve 

 of A'; so our equation must be symmetric Avitli i-egard to the 

 variables ./■; and //,; that is, it has the form 



a%; //,/ -f ;. a\ a, J hh'y = ( 1 ) 



To determine ;. we suppose F to be lying oil ,r^=0 and we then 

 consider the points of h^ which are lying on .i'^=zO. If we represent 

 the linear factors of the binary form «■■'a:=^'^=(«i ■'•i+^s-^'a)^^^ ^W 

 Pj., (Jj: and v'a, then the points H^, H^, H^ are indicated by the 

 equation 



h\-^{p.v q,i + Pii '?.') ipx ry + py rA (q,, Ty + Qy vA — 0, 

 or by 



A'x EÏ ^^ /'':> qx q>i r% + 2 Pj. p,j qj. q,, r^ ry = (2) 



c 



We now ha\e 



3 rt%. ay = Pj- qx Vy -f px q,l Tj: -{- py (/.,• ?V, 



3 h.c h\, ^ Px q>i ry + Py qx I'y + Py qy r,., 

 and as we moreo\er ha\e 



/'// 7'/ ''.'/ = /''>' 

 A\e liiid out of (2) 



h\. = 9a\.ayh,h\^-a\.h'y = (3) 



This equation also represents the harmonic cur\e, if we but again 

 regard a^x ^^ the symbol for {«i.^'i4 «^'''a-f ^'s'^'a) '" • 



2. The polar conic of 1^ with i-egard to I lie curve /', repre- 

 sented by (1) has as equation 



3 rt^, Oy Py + ;. (2 O., a'y h, h' y -j- fl\. Oy !>' y) = 0, 



or, if we put 



a'\ rty EE J^ find «X ^''^1/ zE. -^r 



wc til 1(1 



(3^-;.)/>^,/v + 2;.L^ = o (4) 



It is evident from this that the polar conies of P with respect to 

 the curves of the pencil determined by P and h" touch each other 



