166 Mr Bromwich, "Ignoration of coordinates" 



There are, of course, certain relations amongst the coefficients 

 of the matrix in consequence of the conditions 



P' = P, Q' = R, R' = Q, S' = S. 



In the ordinary optical case (n = 2) these relations have been 

 elaborated by Prof. R. A. Sampson 1 ; and in this case I have used 

 the substitution analogous to " ignoration of coordinates" with the 

 object of shortening the discussion of the optical invariants 2 . More 

 extensive substitutions of a similar character had been previously 

 employed 3 by Prof. Heinrich Bruns of Leipzig, for the purpose 

 of discussing aberration in an optical instrument and other 

 similar problems. 



It is perhaps worthy of remark that the matrix of the last 

 substitution is equal to the product 



o j i\ / i io 



R\ S) K-Q-'Pl Q- 



so that its determinant is 



(- 1)- | R | x | Q-* | = (- 1)» | R | / 1 Q | = (- 1)». 



To illustrate the conclusions arrived at, take the simple case 



V = ax 2 + Ihocy + by' 2 . 



Then the original relations are 



£ = ax + hy, 

 7} = has + by. 



The derived quadratic form is 



U — V — 2y (hx + by) = ax 2 — by 2 ; 



expressing this in terms of x, i), we find 



U t = (a - h 2 /b) x 2 + 2 (h/b) x v - (1/b) v \ 

 so that the derived substitution is 



£ = (a - h 2 /b) x + (h/b) v , 

 -y= (h/b) a: - (1/b) V , 



which has a determinant — (a/b), in agreement with what was 

 proved before. 



1 Proc. Loud. Math. Soc. vol. 29, 1898, p. 33 ; it may be remarked that the 

 relations occur (in a more general form) in connection with some types of contact- 

 transformation. 



2 Ibid. vol. 31, 1899, p. 4. 



3 "Das Eikonal" {Leipziger Abhandlungen, Bd. 21, 1895, p. 325). 



