SATISFYING LINEAR DIFFERENTIAL EQUATIONS. 3 



p r being a polynomial of degree pr, and P r (l/.x) a series of positive integral powers 

 of l/x, the equation has a set of formal solutions of the form 



r = 



where il r is a polynomial of degree p+\, provided a certain determinantal equation 

 has its roots all different. 



The case in which these roots are not all different is discussed by FABRY (' These, 

 Facultd des Sciences, Paris,' 1885), where he introduces the so-called Subnormal 

 Integrals, viz., integrals of the above form in a variable x r ' k , k being a positive integer. 



The investigation carried out in the following bears the same relation to the 

 discussion of these normal and subnormal integrals that Part II. of the paper quoted 

 at the outset bears to the ordinary analysis of the integrals of an equation in the 

 neighbourhood of a point near which all the integrals are regular. 



2. Throughout the discussion the neighbourhood of the point t = will be under 

 consideration, the coefficient p r being supposed to have a pole of order ts r at this 

 point. 



Let p+l be the least integer not less than the greatest of the quantities rav/r. 

 The equation may then be considered as belonging to the more general type 



_ A 



where P r (0 is holomorphic near t = 0. 



This equation may be reduced to a linear system of simultaneous equations as 

 follows (vide ' Proc. Lond. Math. Soc.,' vol. xxxv., p. 344) : 



Put x, = z, o- 2 = ^ +1 z (1> , ... sr r+1 = t r(f+1 W r \ r=l,..,,n-l. 



The n equations then satisfy the system of n equations 





 

 



(n-5 



where Qi...Q n are series of positive integral powers of t. This system belongs to the 

 more general form 



B 2 



