gA KANSAS UNIVERSITY QUARTERLY. 



another conic K' of the range S. Having found the conies K and 

 K' all the rest of the transformation can be constructed. If the 

 conic A' cuts the line 1 in two real points Q and R, then P and 

 P' are outside of K and K' respectively; but if Q and R are a pair 

 of conjugate imaginary points, then P and P' are inside of K and 

 K' respectively; if the conic A' should touch 1, then P and P' are 

 on K and K' respectively. 



The conies K and K' are each characterized by a certain numer- 

 ical constant. Any tangent to the conic K cuts the four fi.Ked tan- 

 events X, V, z, 1 in a constant anharmonic ratio which we shall des- 

 ignate by k. In like manner every tangent to K' cuts the same 

 four tangents in a constant anharmonic ratio which we shall desig- 

 nate bv k'. We shall call these two anharmonic ratios the tan- 

 gcii/ia/ aiilianiioiiii ratios of the conies K and K'. 



Let the conies K and K' touch 1 in the points L and L': let them 

 touch X in X and X', \- in Y antl V. z in Z and Z'. The tan- 

 gential anharmonic ratio k aK)ng the fixed tangent 1 is that of 

 the four points A,, B,, C,, and the point of contact L; thus 

 k=(AjB,C^L); likewise k-^( A, B ,C, L'). Along the line x these 

 same tangential anharmonic ratios are respectively k = (XCBAj) 

 and k' = (X'CBAj). Along z they are k:^(BAZCj) and 



k'=(BAZ'C,). Along y they are k^-(CYAB, ) and k'-(CY'AB,). 



We shall now show that the three characteristic anharmonic 

 ratios A^ , Ay , A^ can be expressed in terms of the two tangential 

 anharmonic ratios k and k . X and X' are corresponding points on 

 the invariant line x: hence A^, -(BCXX'). In like manner 

 Ay=(CAYY') and A, :^(ABZZ'). 



Taking the ranges of points along the line z. we have 



AZ AZ' 



A^^CABZZ')-^ : . Also k:--^(BAZC,): hence 



ZB Z'B 

 I AZ AC, 



-r=(ABZC )=^ — - : : and likewise we have 



k ZB C,B 



AC, AZ' 



k' = (BAZ'C.)=(ABC,Z')rr= : . 



C,B Z'B 

 k' AZ AZ' 



Therefore -^^-•—- : = (ABZZ')-Az. Thus we h.ave 



k ZB Z'B 

 expressed A^ in terms of k and k'. 



Let us next take the ranges ahjug tlie invariant line y. Here we 

 have Ay— (CAYY'_), k=(CYAB,), and k':^-<CY'AB,); whence we 



