1893.] connected tvith BesseVs Functions. 123 



The ordinary expression for J^ in the form of a definite inte- 

 gral is 



Jn 0-r) = , .^'^'Jl T. f '^cos (\r cos 6) sin- ddd. . .(1), 



and accordingly when 7n + 7i is an even integer, the integral VJ" 

 depends upon one of the form 



/•QO 



■^2Sg-aV-cos26^cZ« (2). 



Jo 



The value of this integral when s = is known to be 

 \/7r/2a.e~^^''^'\ from which the value of the integral (2) can be 

 deduced by differentiation with respect to b. 



If however m + n is an odd integer, F„"' depends upon integrals 

 of the form 



/.OO 



1 ic-^+i e~"'^' COS 26^(i^ (3). 



Jo 



This last integral cannot be evaluated in finite terms. 



There is however another form of J^, which is given by the 

 equation 



2 r" 



Jo(Xr) = — / sin (\r cosh ^) (^0 (4), 



and consequently V^'"'^^ depends upon integrals of the form 



f 



Jo 



^.2s+i g-«2x2 gijj 2bxdx (5), 







which can be deduced from the known value of (2) by differentia- 

 ting with respect to b. 



The integral (2) enables F,™ to be integrated with respect to 

 X. when m is an odd integer; but when m is even this integral 

 cannot be employed. Now J^'ir) = — /^(r), and if it were allow- 

 able to differentiate (4) with respect to \r, we should obtain 



J (\r) = 1 cosh 4> cos (X.r cosh 0) dcf> (6). 



vr./o 



We shall hereafter prove equation (6) by a different method ; 

 accordingly when m is an even integer equation (6) enables F,'" 

 to be integrated with respect to X. Since any three Bessel's 

 functions are connected together by the equation 



J^,^(x) = 2nw-W^{x)-J^_^(x) (7), 



it follows that since F„'" and Fj™ can be integrated with respect to 

 A, for any integi'al value of m, the same process can be performed 

 upon VJ\ 



