REGULAR SINGULAR POINTS OF A SYSTEM OF HOMO- 

 GENEOUS LINEAR DIFFERENTIAL EQUATIONS OF 

 THE FIRST ORDER.* 



By Otto Dunkel. 



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



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

 tions of the form : 



^=1 ^ y 



in which the fx.-^/s are constants, and the (i,;/s are functions, not neces- 

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

 interval : 



0< X <b.t 



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

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

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

 multiplied into |«,,y| the resulting function shall be integrable up to the 

 point a: = ; this requirement will be stated more explicitly later.l 



The point x := satisfying these conditions may be called a regidar 

 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, 1002, by the Faculty of Arts and Sciences 

 of Harvard University in f ultilinent of the requirement of a thesis for tlie de-^ree 

 of Doctor of Philosopliy. 



t The requirement that tlic functions n,. .should be continuous in < r < h is 

 made only for tiie sal<e of simplicity. We might aUow tliem to Iiave a finite 

 number of discontinuities in O/^of such a kind that eacli function | <(,. .| can be 

 integrated througiiout the interval ; and ail tiie following work would hold with 

 very little cliange. 



I Cf. p. .'107. 



