REGULAR SINGULAR POINTS OF A SYSTEM OF HOMO- 

 GENEOUS LINEAR DIFFERENTIAL EQUATIONS OF 

 THE FIRST ORDER.* 



By Otto Dunkel. 



Presented by Maxime Bocher, May 14, 1902. Received July 10, 1902. 



We will consider in the present paper a system of n differential equa- 

 tions of the form : 



(i) S = 2(v + °-)^ (.-=i, %...«), 



in which the fx.,- ,-'s are constants, and the a,- t /a are functions, not neces- 

 sarily analytic, of the real independent variable x, continuous in the 

 interval : 



< x < b.t 



We shall require that \a tiJ -\ be integrable up to the point x = 0. 

 For the development of certain sets of solutions we shall make the 

 additional requirement that even after certain powers of log x have been 

 multiplied into | a {J 1 the resulting function shall be integrable up to the 

 point x = ; this requirement will be stated more explicitly later. $ 



The point x = satisfying these conditions may be called a regular 

 singular point of the system of equations (1) in conformity with the use 

 of that term by Professor Bocher in the study of linear differential 



* This paper was accepted in June, 1902, by the Faculty of Arts and Sciences 

 of Harvard University in fulfilment of the requirement of a thesis for the degree 

 of Doctor of Philosophy. 



t The requirement that the functions «-. should be continuous in < x ■< b is 

 made only for the sake of simplicity. We might allow them to have a finite 

 number of discontinuities in 06 of such a kind that each function \a { .| can be 

 integrated throughout the interval ; and all the following work would hold with 

 very little change. 



t Cf. p. 367. 



