as a problem in linear substitutions 
167 
Also, expressing V in terms of x, rj, we have 
V x = (a - h*/b) x* + (1/6) v 2 , 
in which the two sets of variables are separated, according to the 
general result. 
In the corresponding optical transformation (which occurs in 
the theory of a symmetrical instrument) we express y , rj in terms 
of x, f, and find 
V = - (a/h) x + ( 1/h ) £ 
rj = (h — ab/h) x + (b/h) 
This substitution has the determinant — 1 (as found in general), 
but here there is no other relation amongst the coefficients of the 
substitution. 
