(>RK — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 25 



Considering a single discontinuity a;,, such a term is, save as to a constant 

 factor, of the form* 



^-^''^fx-'d^ 



|gM-«.)i?'^(^^c=, -tx)- e-^f^-^'i^^uV, ;U)) 



- W-'^'F.^yce, ii)F,{p^-c\ -fx)- «'""-*)^„(mV, - /x)i?'„(^V, ^,)) j (26) 



This is suitable when the value of x to be considered is less than «,. For 

 X > X,, we would take the expression obtainable from this by interchanging 

 a, b in the numerator. Now, it would evidently establish that the boundary 

 equations at a, and the differential equation (25) are satisfied by (p, Y^, &c., if 

 we could show that initially, for x < «,, (26), its first and second space- 

 derivates, and its first time-derivate, are each zero. This is, in fact, true of it 

 and all its derivates. 



Initially (26) vanishes for all values for which a; <Xi, and therefore so also 

 do all its space-derivates. In order to show that the same is true of its time- 

 derivates, I proceed to examine to some extent the values of (26), and of these 

 derivates when t is small but not zero. These derivates may, as we have seen 

 when t is positive, be obtained by differentiation under the integral sign ; and 

 thus, in the case of any one or of (26) itself, we have, on performing the 

 multiplications indicated in the numerator, to deal with four integrals, each 

 of the type 



{e^'"='*-i-yF(ij.)d,A/'Denomma.tor of (26)), (27) 



oj 



where i^(/u) is an algebraic function, and I - a > y > a - h. Evidently the 

 parts of the contour for which the real parts of ju^ are infinitely great and 

 negative contribute nothing to (27), provided the path avoids the zeros of 

 the denominator. Evidently, also, the remaining path may, when t is less 

 than some finite quantity depending on y, be replaced by two infinite straight 

 lines parallel to the axis of imaginaries and passing through the points 



n = {h-a + y)/2cti, fx^-{b-a- y)l2dh 

 respectively. The integral along the former may be expressed in the form 



- i (c'^-i e-<'-''+!"-/^'^" 



(28) 



l_e-=^(s-a)^(^)' 

 where 0, Z, are algebraic functions involving fractions, and 



u = ifidi - i(b -a + y)/2. 

 It is evident that, if we select any fixed quantity k less than unity, we can 

 choose t so small that, for it and all smaller values of t, the modulus of the 



* Corresponding additions, whose form should lie obvious, must be m.ide to J'.., I's, etc. 

 R.I. A. PROC, VOL. XXIX., bECT. A. [4] 



