The Hon. J. W. Strutt on Bessel's Functions. 329 



used with great advantage. We have 

 i,\ . /'^ ii (l2-4ft2)(3«-4««) , \ / ■rr ■ir\ 



V 



(8.-)^ ■■■>- \ 



2 r P-4n^ _ (P-4»-^) (3^-4)»^) (5^-4n^) 

 irzl 1.8^ 1.2.3(82)3 



+ ...},;„(,_ 5 _„g. (4).* 



If 71 be of the form n= integer -|- ^, the series within brackets 

 terminate, and we are furnished with expressions for J^C-s") in a 

 finite form. For instance, if w = 4, 



^i(^)=A/, 



2 . 



— .sm^". 



But if n be not of this form — for example, if n be integral, — the 

 series run on to infinity, and become ultimately divergent, no 

 matter how large z may be. Nevertheless the convergent part 

 may be usefully employed in calculation ; for it can be proved 

 that the sum of any number of terms differs from the true value 

 of the function by less than the last term included. The most 

 satisfactory demonstration is that of Lipschitz {Crelle, vol. Ivi.), 

 who determines the remainder after any number of terms in the 

 form of a definite integral. 



When 2 is extremely great, J„ reduces to 



J„W=Y/i.cos(^-^-«|), ... (5) 



becoming independent of ?i, except as to its phase. This might 

 have been anticipated from (1). 



The roots of the equations J„(.^) = 0, J'„(<2r) = are all real. 

 When z is very great, they approximate respectively to 



4m 4-271—1 



Z= : TT, 



4m + 2n + l 



Z= :: TT, 



(6) 



* Series (4) was employed by Hansen in his calculations of Bessel's 

 functions. Hansen's Tables will be found in Lommel, Studien iiber die 

 BesseW schen Funktionen, Leipzig. Mr. J. W. L. Glaisher (to whom I am 

 indebted for the loan of Loramel) informs me that the B.A. Committee on 

 Mathematical Tables, of which he is the Secretary, are about to turn their 

 attention to the completion of the existmg Tables of Bessel's functions. 

 Convenient expressions for the calculation of Jn, and the roots of the equa- 

 tion Jn=0, for n—0, and (virtually) n= \, are given in Professor Stokes's 

 paper referred to below. 



