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XL. Notes on BesseVs Functions. By the Hon. J. W. Strutt, 



late Fellow of Trinity College, Cambridge^, 



THE value of BessePs functions is becoming generally recog- 

 nized. By means of them can be solved important pro- 

 blems in mathematical physics relating to the conduction of heat 

 or electricity, the flow of an incompressible frictionless fluid 

 which has been once at rest, and the vibrations of an elastic me- 

 dium, when the nature of the question imposes conditions which 

 are to be satisfied on spherical or cylindrical surfaces. Of late 

 years the development of the subject has been mainly in the 

 hands of German mathematicians. 



In the present paper, which is of a somewhat fragmentary 

 character, are contained some results of more or less novelty, 

 some demonstrations of known theorems by what appear to me 

 to be simple or more instructive methods, together with a few 

 examples of applications to physical problems. 



Different writers have started from difi'erent points in their 

 investigations. So far as concerns its form, the BesseFs func- 

 tion of order n, J„(2'), may be defined as that particular integral 

 of the equation 



dz'^ z dz 



C -?>=«' ••••(!) 



which remains finite when 2'=0. The diff*erential equation 

 gives at once the ascending series 



Z f Z Z 1 



J»W=2.r(n+l) l^~ 2(2/1 + 2) "^2.4, (2^ + 2)(27i + 4)"*""T ^^^ 



with the exception of the arbitrary constant, which is determined 

 by other considerations. 



When n is integral, Jn{^) may be expressed by the definite 

 integral 



J^[z)=~\ cos {zfiin CO — no)) d(o, .... (3) 



1 f- 



--1 



which is BesseFs original form. It readily appears that J„(2r) 

 can never be greater than unity, and, unless z and rt are both 

 small, is always much less. 



The series in (2) never terminates for any value of w, but is 

 always convergent for any value of n or z. However, when z is 

 great, the convergence does not begin for a long time, and the 

 series becomes useless for numerical calculation. In such cases 

 another series proceeding by descending powers of z may be 



* Communicated by the Author. 



