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 y Q' = R t PC = 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 
(° 1 l) x f 1 I <M 
U| S / 1 V- Q~'P | Qr 1 ) ' 
so that its determinant is 
(- 1)“ I R I X I Q- I = (- 1)” I R I / 1 Q I = (- iy\ 
To illustrate the conclusions arrived at, take the simple case 
V = ax 2 + 2 lixy + by 2 . 
Then the original relations are 
f = ax + hy, 
y = hx + by. 
The derived quadratic form is 
U = V — 2y (hx + by) = ax 2 — by 2 ; 
expressing this in terms of x , y, we find 
U 1 = (a — Ti 2 /b) x 2 + 2 (hjb) xy — (1 /b) y 2 , 
so that the derived substitution is 
f = (a — h 2 /b) x + (hjb) y, 
~V= (h/b)x-(l/b)ri, 
which has a determinant — ( a/b ), in agreement with what was 
proved before. 
1 Proc. Lond. 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). 
