SATISFYING LINEAE DIFFERENTIAL EQUATIONS. 11 



ft *T' tt 



-T- = jj+i %, where u is a power series in t, a linear equation of order n and rank p 



near t = can be obtained for each row of the matrix x, and this equation can be 

 solved by the solution of a linear system of the restricted type in question. 



Of the matrix ^ to be used here, the following properties will be presupposed : 



(i.) It is to be of the form % + % + ...+& } where each of the matrices 



V V V 



XI---XP+I has all elements to the left of the diagonal zero. 

 (ii.) The diagonal elements of these matrices are to be numerical constants 



denoted as before by r * (r = 1, ... , p+ 1 ; s = 1, ... , n). 

 (iii.) All the other non-zero elements of x^Xs-'-Xp-t-i are t be constants, while 



the other elements of ^i may contain t, but only to positive integral 



powers (cf. the matrix x in Dr. BAKEE'S paper, loc. cit.). 



8. As before, the matrix 77, which is a formal solution of the subsidiary equation 



drjjdt = UTf)r)x, 

 will be supposed to be formed of the columns 



y = 2/0 



and the equations for the coefficients x r y r ... are the same as the equations X (p. 5). 



But the detailed form of these equations is quite different. The first of them 

 ( ,,+! l p +i)x = is still satisfied by 



3. | I #' n 



~ y i ' ^ p+i ~ PI- 



Supposing now p, to be a root of multiplicity e l} the second equation X is in full 



= = 

 = 0, 



where 



p f ,+i = = p t ^,,^ 



C 2 



