Orr — Extensions of Fourier's and the Bess el- Fourier Theorems. 237 



It X = h, it is readily seen that the integral along the first portion of the 

 contour is 



while that along the second portion is 



If we arrange, as can be done, and as is indeed natural to do, that ^uo = - |Ui, 

 or if, whether we do so or not, 



as is the case in physical examples, the total integral is then 



-«*(J-0j2-^^^j^-jr(r^-y^{. (21) 



the factor in the large bracket being, in the latter case, either or 4, 



If X = a, the integral is, of course, zero. 



Next, consider the integral derivable from (11) by changing the lower 

 limit of u into x, and the upper into h. This integral — and the same is true 

 whatever the range of u — is unaltered by interchanging x and u in the 

 integrand, thereby making it 



'^^J, eMb'a)^,(f,,h)F,(-,,,a)-e^i-~^)F,(-f,,h)F,(f,,a) <^Wclu, 



(22) 

 for the difference between the two integrands is simply 



which integrates to zero when jx describes any closed circuit. 



By reasoning precisely similar to that which precedes, it appears that, if 

 h>x> a, the value of this integral is 



- 2Tri<t>{x + e). (23) 



If x = a, its value, under suppositions similar to those stated in connexion 

 -'*(20). _^,^^,,^^U_^^^^_^^-^)l (24) 



the factor in the large bracket being, in certain cases, either or 4. 



It X = h, the value is zero. 



Thus, by addition, provided Fi, F. are of the same degree as also F^, Fi, 

 we have the equation 



^ r rb {cM(«-«)i^,(-^,rt)-C-'^'«-'')i^,(^,«)} {g"(^-^)J^4(-iU, ?>)-C-^(-^-^)i^s(iU,^)} . . , 



27^cJ '^^Ja" e>^^^--^F,{,,,h)F,{-^,a)-e>^^'^-^)F,{-^,b)F,(f,,a) •^^'"'''' 



= - (p(;x - e) - (j){x + e), a < x < b, (25) 



the integral being taken along a contour everywhere at infinity. 



R.I.A. PROC, VOL. XXVn., SECT. A, [34] 



