DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
159 
gives 
or 
— — ^ (1 + Qw+ Q,uQ,u +...) x° , 
= -^ [l + Q u + Q (uQu) + Q (uQuQu) + ...] cc°, 
= [%+wQw+mQwQw +...] x°, 
— w[l + Qw + QwQw+...]a; 0 , 
= uQ (u) jc°, 
dxjdt = ux, 
so that the functions x — Q ( u)x° satisfy the differential equations. By the definition, 
Qv.ij reduces to zero for t = 0 ; hence 12 ( u) reduces to its first term, the matrix unity, 
when t — 0 ; that is, x — 12 (u) x° reduces to x — x 0 when t — 0. 
In what follows we shall require a particular property of the matrix 12 (u), given in 
‘ Proc. Lond. Math. Soc.,’ XXXV., December 11, 1902, p. 339. If u, v be any two 
matrices of n rows and columns, of similar character to the u considered above, the 
property is expressed by 
12 (u + v ) = 12 (u) 12 {[12 (w)] _1 vQ (w)}. 
where [12 (w)] _1 is the matrix inverse to 12 (u), defined above, such that [12 (w)] -1 12 (u) = 1. 
The theorem is nugatory when the determinant of 12 (u) is zero. It is only equivalent 
to saying that if in the system of linear differential equations 
that is, 
dx 
dt 
(u + v) x, 
dx { 
dt 
{ U il + Dl) X \ + • • • + ( U in + V in) X m 
we introduce a set of n new dependent variables, denoted by 2 , by means of the 
equations 
x = 12 ( u)z , or 2 = [12 (w)] -1 x, 
then 
dz/dt = [12 (w)]^ 1 vQ (u ) 2 . 
This follows at once from 
dx 
( - uJrv)x= f t = s [ q (“) 2] = 11 q (“ ) . 
d 
: + 12 (u) 
dz 
dt 
= \uQ, (w)] 2 + 12 ( u ) = uQ (u) 2+12 (u) ^ 
= ux +12 ( u ) 
dz 
dt 
