336 Lord Rayleigh on Stability of Simple Shearing 

 and imaginary parts we have by (31) from the real part 



f «x e Xv d V . L e- Xn d V -i\ e Xr, d V X 2 e^drj 



- ( Sl e~ X,1 d v . (s 2 e Xrl d v + f *! e ~^d v L e Xv d v = 0, (33) 



and from the imaginary part 



f «i e Xn d v . ft, e' Xv d v + f% <T X ^ . f *, /% 



-fjj! e' Xr) d v . ( t 2 e Xri d v - j 5 2 /% . j ^r X % = 0. 



... (34) 



If we introduce the notation of double integrals, these 

 equations become 



(y s inh\(^-V){*iW^2(V)-*iW .t 2 (< n ')\d v d v , = 0, (35) 



((smh\( V -v'){si(v) .t 2 (v')s 2 { V ) .*i(V)}*?*/ = 0, (36) 



the limits for ?; and ?/ being in both cases ^ and rj 2 . In 

 these we see that the parts for which rj and tj are nearly 

 equal contribute little to the result. 



A case admitting of comparatively simple treatment occurs- 

 when \ is so large that the exponential terms e Xr] , e~ Xri 

 dominate the integrals. As we may see by integration by 

 parts, (31) then reduces to 



S 1 M.S 2 (^ 1 )-S 1 M.S 2 ( % )=0, . . (37) 



or with use of (29) 



S 1 (%).S 1 (^)-r 2 y^=0. "... (38) 



We have already seen that Bi(rj) cannot vanish ; and it 

 only remains to prove that neither can the integral do so. 

 Owing to the character of S l9 only moderate values of rj 

 contribute sensibly to its value. For further examination 

 it conduces to clearness to write 7] 2 = a,7j 1 = — b, where a and 



