SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 31 



If in the linear system 



-| = u (t) y we put t = (j>(z), 



we have without any calculation the new system for y as a function of 2 



Suppose now (f>(z) = z k . Then the transformed system is 



If, then, the original system is of rank p, so that 



the new system is 



dy Vka^_ Jca ,_ 



+ - + - 



and is of rank kp. 



If, now, we were to put z = F' 1 *'' . z 1 , the form of the equation would be unchanged, 

 while the coefficient of z i;!cp+l in the right-hand member would become the original 

 canonical matrix a. p+l . This is, however, not necessary, as the whole investigation 

 could be carried out equally well if any constants whatever replaced the unities to 

 the right of the diagonal in a p+1 . 



It may now well happen that though all the equations of condition found for the 

 general system are not satisfied, those associated with the new system are all satisfied, 

 so that the new system possesses a solution in normal form. If this is so, the original 

 system will be said to admit of a solution in subnormal form. In fact, an integer k 

 can always be found such that this is so, owing to the vanishing of the coefficients of 

 z -*f+r{ r= 0) !...(_ 2)}. 



In the first place, all the conditions arrived at from the equations X,, Y,, ... will 

 be satisfied (p. 13), for the coefficient of z~ kp is identically zero; in general, the 

 left-hand members of X,, Y r . . . are rational integral functions of the elements of the 

 matrices A^, A^-j, ..., A tp _ r+1 , if A, n stands for the matrix multiplying z~'", and 

 contain no term independent of these elements. 



Now the conditions found on p. 13 arise from the equations Xj.-.X,,-!, Y^-.Y^-a, ..., 

 and therefore involve the matrices A^, ..., A A? _ ei+2 . These conditions will therefore 

 all be satisfied if k > ej. Similarly, the analogous conditions for the second group of 

 equal terms in the diagonal of a p+1 will be satisfied if k ^ e 2 , and so on. 



Consider first, as being simplest of explanation and as containing the essential 

 features, the case in which all the roots of the characteristic equation are equal, so 



