Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 19 



value ; and the terms arising from the boundary terms at a due to this partial 

 integration and from the terms involving Jli, combine to give 



c. 



ei^'-'du 



{fxCaU^ - cTAr, „y, d/dx) I e^^'-»>Ft(fic, -fx)- eri^^'=-''^Ft{nc, ^) j ^ . 



-h Denominator of (16) 

 The expression 



/iC nlla - cTAi2 aVl d/dx 



may be expressed as a determinant obtained from nils on multiplying by fic 

 the expressions (13), (14), etc., which constitute the elements of its second 

 column, and subtracting from the first the term cT ai/i d/dx. The determinant 

 thus obtained is unaltered, if from each element of the second column we 

 subtract the sum of the corresponding elements of the first column multiplied 

 by cy„ the third by cy^, &c. And the resulting determinant, which chffers 

 from Afl only in the second column, may be replaced by the sum of A^ and 

 another determinant again differing from Aa in the second column only. On 

 doing so, the part of the integral arising from Aa is zero ; since, if we perform 

 the operation A« on the coefficient of all, in the numerator of (17), we obtain 

 minus the denominator of (17), so that the integrand is devoid of singularities. 

 The part arising from the other determinant, from whose second column the 

 symbolic term containing d/dx has disappeared, has an integrand asymptotically 

 of the form - ei^"' n'^ v-id/i. 



If this were actually the integrand, the integral woiild be iiniformly con- 

 vergent, since the contour might be replaced by a finite one, and its initial 

 value would be v-,. And the integral whose integrand is the difference between 

 this and the accurate expression, having the denominator asymptotically of 

 two dimensions higher than the numerator, is also uniformly convergent, and 

 its initial value is zero. Thus the integral actually gives d/dt („ Y^), and its 

 initial value is Vi. 



Ai;t. 7. This Solution satisfies the Terminal Uqiuitions. 



Before proceeding further it will be convenient to point out that, if in 

 the left-hand members of (7) et seq. we substitute for Y,, Yi, etc., the values 

 An, A,i, etc., the result in the case of (7) is simply Aa and in the case of each 

 of the other equations zero ; while if we substitute alli, nllz etc., the results 

 are respectively Aa x (13), Ao x (14), etc. 



I shall next show that the solution which has been written down satisfies 

 the terminal equations (7) et seq. at a. 



This would be true if the left-hand members of these equations could be 

 legitimately obtained from the values given for f, Y^, etc., by differentiation 



[3*] 



