DIFFERENTIAL EQUATIONS 213 



where G is a constant, then the solution of the differential equation is 



where No and N\ are denned by the expansion, 



a is any root of A(/>) = o, A'(/>) is the first derivative of A(^) with respect to p, 

 and the summation is over all the roots, a. This solution reduces to u = o at 

 t = o. Phil. Mag. 37, p. 407, 1919; Proceedings London Mathematical Society, 

 15, p. 401, 1916. 



9.9 References to Bessel Functions. 



Nielsen: Handbuch der Theorie der Cylinder Funktionen. 

 Leipzig, 1904. 



The notation and definitions given by Nielsen have been adopted in the pres- 

 ent collection of formulae. The only difference is that Nielsen uses an upper 

 index, J n (x), to denote the order, where the more usual custom of writing J n (x) 

 is here employed. In place of Hi n and H 2 n used by Nielsen for the cylinder 

 functions of the third kind, H n l and H n 11 are employed in this collection. 



Gray and Mathews: Treatise on Bessel Functions. 



London, I895. 1 



The Bessel Function of the second kind, Y n (x\ employed by Gray and 

 Mathews is the function 



- Y n (x) + (log 2 - 7)AW, 



2 



of Nielsen. 



Schafheitlin: Die Theorie der Besselschen Funktionen. 



Leipzig, 1908. 



Schafheitlin defines the function of the second kind, F B (*), in the same way 

 as Nielsen, except that its sign is changed. 



NOTE A Treatise on the Theory of Bessel Functions, by G. N. Watson, Cambridge 

 University Press, 1922, has been brought out while this volume is in press. This Treatise gives 

 by far the most complete account of the theory and properties of Bessel Functions that exists, 

 and should become the standard work on the subject with respect to notation. A particularly 

 valuable feature is the Collection of Tables of Bessel Functions at the end of the volume and 

 the Bibliography, giving references to all the important works on the subject. 



9.91 Tables of Legendre, Bessel and allied functions. 

 P n (x) (9.001). 



i A second edition of Gray and Mathews' Treatise, prepared by A. Gray and T . M 

 MacRobert, has been published (1922) while this volume is in press. The notation of the first 

 edition has been altered in some respects. 



