Normal Series satisfying Linear Differential Equations. 339 



" On the Normal Series satisfying Linear Differential Equations." 

 By E. CUNNINGHAM. Communicated by Dr. H. F. BAKER, 

 F.RS. Eeceived December 8, Eead December 15, 1904. 



(Abstract). 



If y denotes a column of n elements, arid u is a square matrix of 

 n rows and columns of elements, each of which is a function of the 

 independent variable, n independent solutions of the system of 

 simultaneous equations 



dyjdt = uy 



are given by the n columns of the matrix 



where Q (<) denotes <j> dt, 

 jt 



In the ' Proceedings of the London Mathematical Society,' vol. 35, 

 pp. 333 ft'., the form of this solution is developed by Dr. H. F. Baker, 

 for the particular case in which each element of u has a pole of 

 order unity at the point t = Q. This is the case of a system derived 

 from a linear ordinary equation of order n, all of whose integrals are 

 regular in the neighbourhood of t = 0. 



If x is any square matrix of order n, and rj a square matrix satisfying 

 the system of equations 



drjjdt = Urj r/X) 

 then y = >/U (x) is a matrix satisfying 



dy/dt = uy. 



This result is used by Dr. Baker in the paper above mentioned to 

 show that in the above case a solution may be found in the form 

 r)ti(x), where the elements of 77 are simply series of powers of t, and 

 the matrix 12(x) is expressible in finite terms. The form of the result 

 puts in evidence what are known as Hamburger's sub-groups of 

 integrals. 



The present paper deals with the linear system of equations in which 

 the matrix u is of the form 



a r , /3 r being matrices of constants. 



The general linear ordinary equation of order n which has t = as 

 a pole of its coefficients is reduced to this form. It is shewn, in the 

 first place, that provided the roots of the determinantal equation 



