330 Mr. A. Cayley on the Homographic Transformation 



Dividing the linear equations by k, and forming with the 

 coefficients on the right-hand side of the equation so obtained 

 a determinant, the value of this determinant is + 1 ; the trans- 

 formation is consequently a proper one. And conversely, what 

 is very important, every proper transformation may be exhibited 

 under the preceding form*. 



Next considering the equations connecting x, y, z, w with 

 f, 97, f, to, we see that 



xf+yf+zf + w 2 ^ f + ^-zif+ao)) 2 

 + (-vt+ v + \£+bco) 2 

 + ( /*f-Xfl; + Z+cco) 2 

 + (-a%-br) - cf+o)) 2 . 

 We are thus led to the discussion (in connexion with the ques- 

 tion of the transformation into itself of the form x 2 -f y 2 + z 2 + w 2 ) 

 of the new form 



( x + vy — fiz + aw) 2 

 + ( — vx+ y + \z + bw) 2 

 -f( fxx — \y-\- z + cw) 2 

 + (—ax—by — cz + w) 2 ; 

 or, as it may also be written, 



(x 2 -f y 2 + z 2 -f iv 2 ) + (vy—fAZ + aw) 2 + (Xz— vx + bw) 2 

 -f- (fix—\y + cw) 2 + {ax + by + cz) 2 . 



Represent for a moment the forms in question by U, V, and 

 consider the surfaces U = 0, V = 0. If we form from this the 

 surface V + qV = 0, and consider the discriminant of the func- 

 tion on the left-hand side, then putting for shortness 



fe=\ 2 + H 2 + v 2 + a 2 + b 2 + c 2 , 

 this discriminant is 



(^Ti 2 +/c^n+0 2 ) 2 , 



which shows that the surfaces intersect in four lines. Suppose 



* The nature of the reasoningby which this is to be established maybe seen 

 by considering the analogous relation for two variables. Suppose that x v y\ 

 are linear functions of a? and y such that a?i 2 4-yi 2 =# 2 +3T; then if 2£=#+#i, 

 2»?=y+yi> i, V w iN be linear functions of a?, y such that £ 2 +r)-=£x-\-yy> 

 or £(£— x) + »?(»?— y)=0 ; £— x must be divisible either by rj or else by rj— y . 

 On the former supposition, calling the quotient v, we have x=z£—vr], and 

 thence y=v£ +J/, which leads to a transformation such as is considered in 

 the text, and is a proper transformation ; the latter supposition leads to an 

 improper transformation. The given transformation, assumed to be proper, 

 exists and cannot be obtained from the second supposition, it must there- 

 fore be obtainable from the first supposition, I. e. it is a transformation 

 which mav be exhibited under a form such as is considered in the text. 



