OF OOicJVJGiiO 



[ 25 ] 



II. 



SOME APPLICATIONS OF BESSEL'S FUIS'CTIOjS'S TO 



PHYSICS. 



By FREDERICK PURSER, M.A. 



Read May 14. Ordered for Publication May 16. Published September 18, lOOG. 



It will be convenient at the outset to investigate certain expansions 

 connected with these functions. 



Retaining the notation which I employed in my previous paper 

 (May, 1902), I emi^loy further the symbol Yq to denote a second 

 integral of the differential equation 



dx- X dx j 

 satisfied by u = Kq. Yq may then be defined by 



This may be written in the form 



Yq{x) = Kq{x) . log^r + iKlx) + <l>{x), 



e being a certain numerical coefficient, <^(^) an even function of x, 

 viz. dox"^ + a^x^ -f . . . , the coefficients of which are determined by 



Substituting for Ki{x) its value, we have the following determination 

 for the coefficients cJo, <^4j (^c 



1 3 11 



a convergent series. 



Yq{x) being thus defined, Yy[x) is given by Yy{x) = 

 will be observed that for all positive values of .r, YJx) is negative. 



U.I. A. PKOC, VOL. XXVI., .SEC. A.l C 



