3-42 PROCEEDINGS OP THE AMERICAN ACADEMY. 



equations of the second order. * That this terminology is a legitimate 

 extension of that commonly used when the coefficients of the system of 

 differential equations are analytic functions of a complex variable, will be 

 evident if the results of the present paper are compared with the thesis 

 by Sauvage : Theorie generate des systemes d 'equations differentielhs 

 lineaires et homogenes.'f 



Our object is to investigate the nature of the solutions of (1) in the 

 neighborhood of the regular singular point x = ; and for this purpose 

 we shall first reduce the system of equations to a canonical form by 

 means of a linear transformation with constant coefficients of the de- 

 pendent variable. We shall then apply the method of successive 

 approximations to develop about the point x = Oa system of n linearly 

 independent solutions of the canonical system. By means of the linear 

 transformation we shall return to n linearly independent solutions of the 

 original system ; and finally an application will be made to the case of 

 the single homogeneous linear differential equation of the nth order. 



§1. 



A Special System of Equations: Its Reduction to a 

 Canonical. Form, and Solution. % 



Let us first examine the special case of (1) in which the coefficients 

 a t j are all zero. In this case we have the system of differential 

 equations : 



(3) 1=2^ (* = 1,2,... K ), 



3 =\ 



where the ju.,,yS are constants. 



A solution of this system may be obtained in the following way. 

 Substitute 



y. =z C t x r Ci = constant 



* Cf. Trans. Am. Math. Soc, Vol. I. Jan. 1900, p. 41. The results of this paper 

 are included as a special case in those we now give. Cf. § 7. 



t Paris, 1895. Reprinted from the Annales de la Faculte' des Sciences de 

 Toulouse, Vols. VIII. and IX. 



| The results of this section are not new, being on the one hand only slightly 

 modified forms of Weierstrass's results (cf. the foot-note on p. 345), and on the 

 other hand special cases of the results obtained by Sauvage (cf. the last foot-note.) 



