328 Mr. A. Cayley on the Nomographic Transformation 



In fact, these values satisfy identically the equations 



=0, 



i. e. the point {x^ y 2i z 2i w 2 ) will be a point in the line joining 

 Oi> V\y *u w \) and ( a > & Y> S)- Moreover, 



(a . . )(#2, y 2 , * 2 , w 2 ) 2 = (a . . Kfn ifu *i> V\Y 



- y ( fl • -)(*i A % $)(*i> Vi> *\> w \) 



that is, 



(a . . )(# 2 , y„ s 2 , w 2 ) 2 = (« . . ){x v y v z v w x f. 



So that x v y v z v w x being a point on the surface, X& y 2 , z 2 , w 2 

 will be so too. The equation just found may be considered as 

 expressing that the linear equations are a transformation of the 

 quadratic form (a . . ){x i y,z i w) 2 into itself. If in the system 

 of linear equations the coefficients on the right-hand side were 

 arranged square-wise, and the determinant formed by these 

 quantities calculated, it would be found that the value of this 

 determinant is — 1 . The transformation is on this account said 

 to be improper. If in a system of linear equations for the trans- 

 formation of the form into itself the determinant (which is 

 necessarily +1. or else —1) be +1, the transformation is in 

 this case said to be proper. 



We have next to investigate the theory of the proper trans- 

 formations of a quadratic form of four indeterminates into itself. 

 This might be done for the absolutely general form by means of 

 the theory recently established by M. Hermite, but it will be 

 sufficient for the present purpose to consider the system of equa- 

 tions for the transformation of the form x 2 + y 2 + z 2 -f w 2 into 

 itself given by me some years since. (Crelle, vol. xxxii. p. 119.*) 



I proceed to establish (by M. Hermite's method) the formulae 

 for the particular case in question. The thing required is to find 



.* It is a singular instance of the way in which different theories con- 

 nect themselves together, that the formula; in question were generalizations 

 of Euler's formulae for the rotation of a solid body, and which are formulae 

 which reappear in the theory of quaternions ; the general formula; cannot 

 be established by any obvious generalization of the theory of quaternions. 



