1G0 
PROF. H. F. BAKER ON CERTAIN LINEAR 
which gives 
Q (u) — vx = 
v ' dt 
vQ ( u) z. 
In what follows we shall generally write hff 1 (u) in place of [Q (w)] _1 . 
Another property to be noticed* is that the determinant of the matrix 0 (it) is 
equal to the exponential of the sum of the integrals from 0 to £ of the diagonal 
elements of the matrix u. For, if il %j denote the general element of H (it), the 
equation 
Q (it) = uQ (it), 
already remarked, is the aggregate of the equations 
— U il®lj + ••• + U in& 
ny 
Further, the differential coefficient of a determinant of n rows and columns can be 
written as a sum of n determinants, each of which is obtained from the original 
determinant by replacing the elements of one row respectively by their differential 
coefficients. Hence we at once see that, if A denote the determinant of £1 (u), 
dA/dt = (u n + u. 22 + ... + u nn ) A, 
which establishes the result in question. 
In particular, if the sum of the diagonal elements of u, 
Un + U 2 2 + •.. + u nn , 
be zero, the determinant of Q(u) is independent of t, and is thus equal to unity. 
This result is of frequent application. 
§ 13. After these introductory remarks we may at once show that the equation 
y-72/y, 
—' + x (l + 4a cos ht + 46 cos Jet) = 0, 
Civ 
to which reference has been made, is capable of solution as an absolutely and 
uniformly converging series in a , b, whatever h and Jc may be. It will be as simple, 
and of utility for other applications we wish to make, to take the equation 
i^ + x(n 2 + dr) = 0, 
in which we may suppose n to be an integer. 
* Cf. Darboux, ‘ Compt. Rend.,’ XC. (188.0), p. 526. 
