2 MR. R CUNNINGHAM ON THE NORMAL SERIES 



where u is a square matrix of n rows and columns whose elements are functions 

 of t, and x denotes a column of n independent variables. 



A symbolic solution of this system is there given and denoted by the symbol fl(u). 

 This is a matrix of n rows and columns formed from u as follows : Q(<) is the matrix 

 of which each element is the (-integral from t to t of the corresponding element of <, 

 (j) being any matrix of n rows and columns ; then 



wQw...ad inf., 



where the operator Q affects the whole of the part following it in any term. 

 Each column of this matrix n(n) gives a set of solutions of the equations 



dx/dt = ux, 



and since fl(n) = 1 for t = t , these n sets are linearly independent, so that fl(u) may 

 be considered as a complete solution of the system. 



Part II. of the same paper discusses the form of the matrix Sl(u) in the neighbour- 

 hood of a point at which the elements of the matrix u have poles of the first order, 

 or in the neighbourhood of which the integrals of the original equation are all 

 " regular." 



It is there shown that if t = be such a point, a matrix 



0* c,. 2 (t/t ) e - 



0. 



can be found, in which all elements to the left of the diagonal are zero, in which 

 c^ = unless t Q } is zero or a positive integer, such that fl(u) is of the form 



where G is a matrix whose elements are converging power series in t, and G is the 

 value of G at t = t a . 



The form of <f> is such as to put in evidence what are known as HAMBURGER'S sub- 

 groups of integrals associated with the fundamental equation of the singularity ; the 

 method is, in fact, a means of analysing the matrix fl (?t) into a product of matrices, 

 of which one is expressible in finite terms and shows the nature of the point as a 

 singularity of the solution. 



The object of the following investigation is to see how far, under what conditions, 

 and in what form, such an analysis can be effected for equations having poles of a 

 higher order than unity in the elements of the matrix u. 



It is known that if in the neighbourhood of infinity the equation is of the form 



X 



