A THEOREM ON DIFFERENTIAL FUNCTIONS 



By S. Epsteen 



In the theory of Algebraic Equations^ it is shown that every rational 

 symmetric function of the roots of an algebraic equation can be expressed 

 rationally in terms of the coefficients. As a corollary of this Professor 

 James Pierpont proves in Part I of his lectures on "Galois Theory of 

 Algebraic Equations:"' Let /? be a given domain of rationality and 



}(x) =x^+aiX"-^ + a2X^~' + a3X^~3 -\- .... an-iX+a„=o , 



an equation whose coejBScients lie in R. Let its roots be |,, |j, . . . . , |„. 

 Then an integral rational symmetric function of « — i of the roots, say 

 ^2, ^3, . . . . , ^„, is an integral rational function of the remaining root l^. 

 In the Theory of Linear Differential Equations it is shown^ that a 

 rational symmetric differential function of the integrals of a fundamental 

 system y^ . . . . y„ can be expressed rationally in terms of the coeffi- 

 cients and their derivatives. Let Rhea given domain of rationality and 



„ d^y d"~^y dy 



'^y=d+^'d^'+ ■■■■ +^«-i+^"^=°> 



a linear homogeneous differential equation with coefficients in R. 

 Then as a corollary of Appell's Theorem we show (in analogy with the 

 Theory of Algebraic Equations) that: An integral rational symmetric 

 differential function of (n—i) solutions, say y^ . . . .yn, is an integral 



rational function of ■)]=— and its derivatives, y^ being the remaining 



solution. 



dy 

 The differential equation Ly = ~ — r)y^o has for a solution y=yj. 



» BuRNSiDE AND Panton, Vol. I, Art. 78. (FouTth edition, 1899.) 



' Annals of Mathematics, 2d Series, Vol. I, No. 4 (1899-1900), p. 116, Art. 3. 



3 Appell, Annates de VEcole Normale Superieure, Series II, Vol. X (i88i), p. 400. See Schleslnger, 

 ■ Handbuch der Theorie der linearen Differenlialgleichungen, Vol. I, Part i, p. 41 



127 



