M.-P.-Vol. I.] HASKELL— QUADRATIC TRANSFORMATIONS. 3 



( 7 ) Q-i; a- \ = r 2 r 2 , cr /* = r 3 r, , <r v = r, r 2 



(8) 6*!- 1 : t x 2 = L 2 \ + M z p + TV, v 

 r # 3 = Z 3 X -1- J/ 3 /* + N 3 v 



Compounding these formulas, the inverse transformation 

 may be conveniently written: 



P * = z, r 2 r 3 + m, r 3 ^ + iv; r x r 2 =^> (yi,yt,y*) 



( 9 ) P * 2 = z 2 r 2 r 3 + j/ 8 r 3 r x + jv a n r 2 =v ( yi , y , ,y a ) 

 P x 3 = u r 2 r 3 + Ms r 3 r, + yv 3 ri r 2 =x o*,^,yO 



The above inversion is of course sufficiently simple, but 

 I have preferred to write out the formulae in detail, as some 

 of the results will be convenient for comparison in the in- 

 vestigation of transformations in the general form which is 

 to follow. 



§ 2. Geometrical theorems and constructions. 



Before proceeding to the problem of the inversion of 

 transformations given in the general form, I will point out 

 that the formulas of § 1 lead to a theorem which seems to me 

 to be a fundamental theorem in the geometrical theory of 

 quadratic transformations. 



From (7) we have 



(10) \ : v = Y% : Y\ and ^ : v = Y 3 : Y%. 



Hence, to the intersection of 



(11) \ — k v = o and fi — k' v = o 



corresponds the intersection of 



(i 2 ) Y 3 — kY x = o and Y 3 — k' Y* = o, 



where k and k' may have any values whatever. But 

 X — k v = o and Y 3 — k Y\ = o represent for variable 

 k two homographic pencils, while /* — k' v = o and 

 Yz — k' Y 2 = o represent a second pair of homographic 

 pencils. A slight generalization of these formulas leads us 

 to the following: 



( 2 ) January 14, 1898. 



