342 The Hon. J. W, Strutt on BesseVs Functions. 



The first value of jo is zero ; but we have already included the 

 corresponding terms in (f>Q. The next values are 



3-8318 13-324 



7*0152 16-471 



10-1735 19-616, 



approximating to the form 



(m-fi)7r--151982 



TT 



■)f 



When z=0, 



4m +1 



-P = -Sp«^Jo(;?r); 



dz ~ ''^ ^ 

 so that the condition to be satisfied from r=0 to r=l is 



Multiplying both sides by ^oipr) and integrating from to 1^, 

 every term but one on the left vanishes, and v^e have by the 

 theorem 



Jo 

 or, since from (1) 



J"o(;^)+Jo(;')=o, 



Now, on the right-hand side, 



Ci 1 /> _ 1 fP 



I rdrJQ{pr) = -^\ a;dxJ ^{w) = -^ \ ^<r{.rJV<^)-f JV^)| 



= ^[i'J'o(^)]=0. 



* See a paper by Professor Stokes in the Cambridge Transactions, 

 vol. ix., " On the Numerical Calculation of Definite Integrals and Infinite 



Series." In example III. the integral I xdxJo(x) is considered under the 



Jo 



notation v. Now, since J"o-f - J'^-f Jo=0, it follows that 



i 



xdxJf){x) = — xJ'(^{x) = a? Ji(a?) ; 



so that the roots of the equation v = 0, given by Stokes, are none other 

 than the roots of J'o(«) = 0, or Ji(a?)=0. 



