as a problem in linear substitutions 167 



Also, expressing V in terms of x, t], we have 



V 1 = (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 



y = — (a/A) x + (l/h) g, 

 v = (h- abjh) x + (b/h) f. 



This substitution has the determinant — 1 (as found in general), 

 but here there is no other relation amongst the coefficients of the 

 substitution. 



