62 Proceedings of the Royal Irish Academy. 



or, if we retain only terms of the first order of small quantities 

 d 



clt^ 



rdel \dr' r dr r^ dd'J dr \r dr ^' ^\ ''' ^^^ 



with the condition that xjj is to vanish at the boundaries. 



If we were now to make i^ as a function of B and t vary as e«('»^+*^), it 

 might be shown that the equation in n, to which the boundary conditions 

 lead, cannot be satisfied by a complex value of n, if d/dr {r~^d{r V)/dr) is one- 

 signed throughout, which is evidently Lord Eayleigh's argument alluded to. 



Whether we make this particular supposition or retain (3) in its most 

 general form, it is evident that the equation is intractable, unless the velocity 

 in the steady motion is such that 



dr \r dr ) ' 



or V=Cr+C'r-\ (4) 



This law, however, is that which applies in the case possessing the chief 

 physical interest as being that which holds for viscous fluid when one or 

 both of the bounding cylinders are made to rotate. 



Taking this law, then, and supposing that \p varies as 6«(''<+«^), equation (3) 



becomes 



Vs\fd'-^ 1 dip s- \ ^ ^., 



n + — -r-T + - ^ ^ = 0. (5) 



r J\dr^ r dr r^^J ^ ' 



The solution of this, subject to the given boundary conditions, resembles that 

 of the preceding problems, in that it involves slipping in the interior of the 

 fluid. If the outer and inner radii are a, h, it is 



^ = A {r'h-' -r-'h') 

 throughout a region adjoining the inner boundary, and 



ip = B {r^a'^ - r~^ a^) 

 through a region adjoining the outer, the surface of separation being that for 

 which w -f s Vjr is zero, and the coefficients A, B being so connected that the 

 value of ■<\i is continuous. And it may again be proved that any disturbance 

 of an ordinary type can be resolved into elements, each one of which is as 

 described, by aid of the equation 

 'As{a?l''-a-'}f)f[r) 



= [^a-^-r-^a^) \\p'h-^ - p-'hy,pr[p) +f{p)- s^p-f{p)]df, 



+ {r^lr^ — r *5* 



[p^a^ - p-W) { prip) +f(p) - sYfip) ] dp, (6) 



provided f(a), f{b) are zero, and /(r), f'{r) are finite, continuous, and diffe- 

 rentiable throughout the region. This equation may be discovered as before, 

 and is easily verified. 



