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, 77 it becomes U x , then its matrix is 
known to be 
C'BC = A, 
and we have thus the rule given by Routh (for dynamical cases) 
p _1 dUi _ 1 dU 1 /r = 1, 2, ..., m\ 
_ 2 dx r ’ dv,’ U = 1 , 2, .... n ) ' 
We may remark that the determinant of U 1 is 
\A\ = \B\x\C,\ 
now | ^ | = | P | x l | = (- 1WP | x | 5 J , : \C\^ \ S~' | = \S\~\ 
so that | ; A |' = (— l)”"f P | / j $ | . 
A result equivalent to the last was given in Part I. of the Mathe- 
matical Tripos, 1898. 
Suppose that instead of applying the substitution C to U, we 
applied it to V ; the resulting quadratic form in x, 77 would be, 
say, V lt and its matrix would be 
1 
CQ 
1 
1 — 1 
) x ( p 1 Q\ x 1 
( 1 I O' 
\ [P-QS-'R | 0 ^ 
Vo | s- 1 . 
J \B | SJ ' 
v- S-'R | 
/ _ V 0 1 8-y 
It follows that in V x the two sets of variables x, 77 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 77 ’s in terms of the os’s and £’s. Following 
a process similar to that adopted above, we find (if | Q \ =}= 0 ) 
y = - Q-'Px + Q- l t 
77 = (R- SQr'P) * + SQr% 
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 
