Ok It — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 31 



be taken initially. Integrate this integrand alone along a path consisting of, 

 (i) a closed contour surrounding the origin and everywhere at infinity, 

 (ii) a contour liaving the same beginning and end as (i), but enclosing one and 

 only one of the branch-points, (iii) the contour (i) again, (iv) the contour (ii) 

 reversed. This will give the same result ; for in (ii) the final value of the 

 radical is equal in magnitude and opposite in sign to the initial, so that the 

 contours (i) and (iii) give the previous results and the integrals along (ii) and 

 (iv) cancel each other. 



In reducing the preceding integrals to sums of residues we may first 

 draw small contours round each of the branch-points, and then join each of 

 these contours to some common point in the plane by curves which do not 

 pass through any zero of the denominator. The sum of the two terms in 

 (46), for example, reduces to the sums of the residues arising from both 

 terms due to all zeros of the denominators otiier than the branch-points, 

 taken along with the integrals of both terms along the small contours 

 surrounding the branch-points ; for along the curves joining these contours 

 to the common point the integral from the first term and that from the 

 second cancel each other. The integrals alontj the small contours are zero 

 unless the branch-points themselves are roots of the characteristic equation. 



Art. 14. A Circular or Annular Memhrane, itself siibject to Viscous Forces. 



We may similarly modify the problem of § 12 by introducing viscous 



forces in the lamina itself. In that case (34) is replaced by 



{d'jdr- + r-'djdr - n-r^) (c> + gd,l>ldt) = d-(^ldt'' +fdfldt, (49) 



and type-solutions are 



^ = e^'IC,, (± fir), 



where the equation connecting /a, v is identical with (35). 



The solution may be obtained by modifying that of § 1 2 in a fashion 

 similar to that in which the solvxtion of § 5 has been modified to give that of 

 § 13 : it seems unnecessary to actually give it. 



Art. 15. The above Solutions agree loith those f omul hy aid of the characteristic 

 Property of Normal Functions. 



Eeturning to the problem of Arts. 5-8, the form of equation (16) suggests 

 the following theorem : — If ^, Y, denote the displacements in any one funda- 

 mental motion having the time-factor ei^"*, and if the same letters with dashes 

 denote the corresponding quantities in any other fundamental mode, then 



Tii^ ^ iu'. 



<t,<p'dx + „n, [fx., Y\) <i,alA,, (m) + jH, [fx, Y'i)i,i/Bu(fi) = 0, (50) 



