M.-P— Vol. I.] HASKELL— QUADRATIC TRANSFORMATIONS. II 



X = o of the inverse transformation are at most linear 

 functions of the conies of (22). We shall see that they are 

 identical with the latter. 



§ 5. Identification of the equations, just obtained zvith 

 those of the inverse transformation. 



From the form of equations (22), we see that the first of 

 these equations is also satisfied by the points b and c, the 

 second by c and a, the third by a and b. But these are 

 exactly the conditions which were to be satisfied (§ 3) by 

 the curves <& = o , M/ 1 = o , X = o. 



The left-hand members of equations (22) are therefore, 

 except for eventual numerical factors, identical with the 

 forms <J>, >?, X of the inverse transformation. 



These factors shall now be determined. Let 



p *i = & l(y ° O 2 — 4 (y b f) (yf c )] 



(23) px 2 = k 2 [(yea) 2 — 4 (yeg) (yga)] 

 p x 3 = k 3 \_(y a b) 2 — 4 (yah) (y h b )] 



The consideration of a single pair of corresponding points 

 will be sufficient to determine the ratios of k\ : k 2 : k 3 . Let 

 us write ji : y 2 : yi = 1 : o : o, when 



p xi=k\ \( b 2 c 3 — b 3 c 2 ) 2 — 4 ( b 2 f — b 3 f) (f 2 Ci—fc 2 )^\ 



(24) p x 2 =k 2 [(c 2 a 3 — c 3 a 2 y — 4 (c 2 g 3 — c 3 g 2 ) (g 2 a 3 — gza 2 )~\ 

 p x 3 =k 3 \_(a 2 b 3 — a 3 b- 2 y — 4 (a 2 h 3 — a 3 h 2 ) (h 2 b 3 — h 3 b 2 )~\ 



which can be reduced by substituting from (16) into 



pXl = —k l L l Mi JV t ( U Qi jRi + M\ R\ Pi + 

 Nx Pi Qi ) 



(25) P x 2 = — k 2 L 2 M 2 N 2 ( L 2 Qi Ri + M 2 R x P x + 



A 2 Pi Qi ) 

 P x 3 = — k 3 L 3 M 3 N 3 ( Z 3 Qi Pi + M 3 Ri Pi + 

 N 3 Pi Qi ) 



But, substituting y t : y 2 : y 3 = 1 : o : o in formulae (9), 

 the equivalent of (23), and comparing the result with (25), 

 we find that 



(26) h Li Mi Ni = k 2 L 2 M 2 N 2 = h L 3 M 3 N 3 . 



(2) January 31, 1898. 



