ME. E. CUNNINGHAM ON THE NORMAL SERIES 





where a p+1 ...ft... are square matrices of constants. 



The most general equation of this form will be considered. 



If p. be any matrix of constants and y = px, the n quantities y satisfy the system 



a 

 or 



Let /i be now chosen so that (p^fT 1 ) is of canonical form as follows :- (i.) It has 

 zero everywhere save in the diagonal and the n-1 places immediately to the right of 

 it ; (ii.) The diagonal consists of the roots of the equation a p+l -p\= 0, equal roots 

 occupying consecutive places; (iii.) The elements to the right of the diagonal consist 

 of (e,-l) unities, then a zero, (e 3 -l) unities, a zero, and so on (' Proc. Lond. Math. 

 Soc.,' vol. xxxv., p. 352). 



Form now the matrices (/lo^' 1 ) ...(/tftfT 1 ) ; the equation is then replaced by an 

 equation of exactly similar form, the matrices a,,... being still any matrices whatever, 

 but Oj, +1 being of the canonical form. 



3. The equation being denoted by 



dy/dt = uy, 



if 77 be any solution of the equation 



(A) drjjdt = U7)-r)x, 



X being an arbitrary matrix, we have 



so that i/n (x) is a matrix satisfying the equation in question. 



In what follows we are concerned with the form of a solution more than the actual 

 convergence and existence of the same. It is therefore important to notice that if 77 

 be a diverging power series formally satisfying equation (A), 770 (\) may be still 

 considered as a formal solution of the original equation, the only condition necessary 

 to secure its actual existence being the convergence of 77. 



If 77 be convergent, the solution may be particularized by adding the factor ij ~ l t 

 i.e., 170 (x)^)" 1 i s the solution reducing to unity at t = t a . 



The main investigation to be carried out is that of a simple form for the matrix x, 

 such that the subsidiary equation (A) may have a formal solution in the form of a 



