342 PROCEEDINGS OF 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 a? = ; 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 = a system of 7i linearly 

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

 transformation we shall return to fi linearly independent solutions of the 

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

 the siuo-le homogeneous linear diffijrential 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, J are all zero. In this case we have the system of differential 

 equations : 



(3) i=2^'^^ (.•=l,2,...n), 



where the /jt,,yS are constants. 



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

 Substitute 



y. = dx'' Gi = 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. 



t 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 

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



