158 
PROF. H. F. BAKER ON CERTAIN EINEAR 
Thus we see that each of the n 2 infinite series constituting the elements of the 
CD 
matrix 
0 ( 'u ) = 1 t C^u T - QwQw +... 
has terms whose moduli are respectively equal to, or less than, the real positive terms 
of the corresponding infinite series constituting the elements of the matrix 
l+sM+Vyp+V M 3 +.. 
This last is, however, certainly convergent for all finite values of s, whatever be 
the (finite) values of the elements of the matrix M. For the case when the algebraic 
equation satisfied by M has unequal roots, its sum is given by the formula of ‘ Proc. 
Lond. Math. Soc.,’ XXXIV., February 14, 1901, p. 114, which can be easily modified 
to meet the case of unequal l oots. 
Thus each of the elements of the matrix Q (u) is an absolutely and uniformly 
convergent series ; in the case when the elements u {j are functions of the complex 
variable, as explained above, it follows that every element of the matrix Q ( u ) is a 
function of the complex variable, and differentiation (and integration) of the series 
representing this elements is permissible, term by term. For the case of real functions 
we introduce this as a condition. 
Hence, if cc° denote a row of n arbitrary values aq°, x 2 °, ..., x n °, the row of n 
quantities denoted by 
x — Q (u) x° 
is at once seen to form a set of n integrals of the differential equations, reducing for 
t — 0 to the arbitrary values x°, that is, x { reducing to x°. For if v denote any 
matrix, of n rows and columns, whose elements are differentiable functions of t, if x° 
denote a row of n constants, and y the set of n functions given by 
that is, 
we have 
y = vx°, 
Vi — V il X l + V x2 X 2 + • • • + V in X n> 
,d± 
dt dt 1 dt 
in i) 
tX/ -» 1 
which, if 4^ denote the matrix whose elements are the differential coefficients of the 
dt 
elements of v, we can denote by 
dy dv o 
dt dt ' 
x = 12 (u) x°, 
Hence the equation 
