Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 235 



another. It is supposed that Fi , F^ are of the same order, that they do not 

 vanish for a common value of jx, and that the same is true of F3, F^. I 

 suppose I > a, but make no supposition as to the sign of either : in some 

 physical examples a is zero, in others, a = - h. The solution which follows is 

 almost obvious from the work of Carslaw. 



I suppose that the function to be expanded satisfies Dirichlet's conditions. 



Each term of the sum is a multiple of 



e" (^ - «) 1; (- ju, a) - e-i" (^ - <^) F, (^, a) , (7) 



and at the same time a multiple of 



gMx- J)i?;(_ ^^ i) _ e-/x(x-5)iP3(^^ J), (8) 



The equation which determines the admissible values of fx is 



e>^^'-)F,{~,x,a)F,(fx,h) - e^^-^)F,{- fx,l)F,{n,a) = 0; (9) 



this equation evidently has an infinite number of roots, and those whose 

 numerical value is large ultimately tend to the form 



fi = mri/{h - a) + a, (10) 



where n is a large integer, positive or negative, and a is some finite constant 

 which in physical instances is generally a pure imaginary. 

 Consider the integral 



^^ ( e^ ^^^-^)F.{-(Ji,a)-e-^ ("-°)i^i(^,ft) } { e^ ^^-^F,{-fx, h)-^'^<-^-^F,(fi, h) ] 



c^ (*-) F^ijx, h)F,{-ix, a)-ey- ^-'^F,(-fx, b)F,{,x, a) ^ ^''^ '^''' 



(11) 

 where the path of ^u is a closed contour everywhere at a very great distance 



from the origin, and which does not pass through any zero of the denominator. 



The path may be supposed to pass half way between the last zeroes included 



and the adjacent zeroes first excluded, 



I first suppose x <!). 



First, consider the portion of the contour to the right of the axis of 

 imaginaries. Along this portion, wherever arg. ^ differs from + 7r/2 by a 

 finite quantity, the most important term in the numerator of the fraction in 

 the integrand is 



- eM(«-a)i^^(_ ^,^ a) . e-'-(^-*)i^3(M. ^); (12) 



in the denominator the former term is the more important. Thus, except for 

 the arguments ± 7r/2, the fraction is asymptotically equal to 



_gM{«-x)^ (13) 



Now, if we substituted this asymptotic value for the fraction, the integral 

 along this portion of the contour would be 



-0(.;-e)log,(^,/;.O,* (14) 



* See Part I., Art. 20. "With the whole investigation compare Picard, l. c. 



