as a problem in linear substitutions. 165 



it is then easy to see that the matrix of the coefficients of U 

 is the matrix denoted above by B. Suppose that when U is 

 expressed in terms of x, rj it becomes U 1 , then its matrix is 

 known to be 



C'BG=A, 



and we have thus the rule given by Routh (for dynamical cases) 



1 dUi _ _ldU l fr = l,2,...,m 



f '-~2 dx/ y °~2 d Vs ' \s = l,2, ...,n 



We may remark that the determinant of U 1 is 



| A | = | B\ x | C, a , 



now \B\ = \P\x\-S\ = {-iy\P\x\S\, \C\ = \S- X \ = \S\~\ 



so that \A\ = (-l) n \P\l\S\. 



A result equivalent to the last was given in Part I. of the Mathe- 

 matical Tripos, 1898. 



Suppose that instead of applying the substitution G to U, we 

 applied it to V; the resulting quadratic form in x, 77 would be, 

 say, V l , and its matrix would be 



1 | - QS-^ „{P\Q\ ( 1 0^_ (P-QS-'R | 



o | s- 1 J \R\sj \- s-^R | s-y V o \s 



It follows that in V x the two sets of variables x, rj are separated ; 

 a result which has important consequences in the theory of re- 

 ducing quadratic forms 1 and in dynamics 2 . 



A somewhat different transformation occurs in Optics, when 

 deducing the equations of a ray of a thin pencil (after passage 

 through any optical instrument) from the properties of the 

 characteristic function. In this case m = n, and it is necessary 

 to express the y's and r/'s in terms of the x's and £'s. Following 

 a process similar to that adopted above, we find (if | Q \ =}= 0) 



y = - Q-iPx + Q-% 



v = (R - SQ-iP) x + SQ-% 



but I do not know of any means to bring the matrix of this sub- 

 stitution into a symmetrical form. 



1 Cf. a paper to appear in the July number of the American Journal of Mathe- 

 matics. 



2 For instance, Thomson's (Kelvin's) and Bertrand's theorems on systems 

 started from rest by impulses can be readily deduced from this. 



13—2 



