Flow of Viscous Liquids. 369 



in which the constants A , B , A 1? . . . arc to be determined 

 from the conditions that yfr and d-fyjdy vanish when 

 y = B cos k.v. It may be observed that the problem is 

 mathematically identical with that of an elastic plate clamped 

 at a sinuous edge, and deformed in such a manner that if 

 there were no sinuosity the bending would be one-dimen- 

 sional. 



The boundary conditions are — 



Ao + B /3 cos kx + L/3 2 cos 2 kx 



.+ cos kx ( A 1 + B x /3 cos kx) er* cos kx 



+ cos 2kx (A 2 + B 2 /3 cos hx) e~ 2 ^ coakx 



+ =0 (48) 



and 



B + 2L/3 cos kx 



+ cos kx (B x — kA l — Bik/3 cos kx) e~ k ^ coskx 



+ cos 2kx (B 2 -2AA 2 -2B 2 k/3 cos kx)e-^ OMhB 



+ = 0; (49) 



or, with use of (48), 



kA + B + {B kj3 + 2L/3) cos kx + Lk/3 2 cos' 2 kx 



+ B l coskxe~ klBc03lcx 



+ (B 2 -kA 2 ~B 2 k/3 cos kx)e-^ coakx 



4- = (50) 



The exponentials in (48), (50) could be expanded in 

 Fourier's series by means of BessePs functions of an ima- 

 ginary argument, and the complete equations formed which 

 express the evanescence of the various Fourier terms. But 

 the results are too complicated to be useful in the general 

 case ; and, if we regard k/3 as small, it is hardly worth while 

 to introduce the Bezel's functions at all. The first approxi- 

 mation, in which /3 2 is neglected in (48), (50), gives 



A = 0, A, = 0, A 3 = 0,...-1 



B =0, B 1 =-2L / 8, B 3 = 0,..J' • ' W 



and the second approximation, in which ft 2 is retained, gives 



A = i/3 2 L, A 1 = 0, A 2 = i/3 2 L,.... 1 



B =-2&/3 2 L, B t =-2/3L, B 2 = -&/3 2 L, . . . . J ' } 



the coefficients with higher suffixes than 2 vanishing to this 

 order of approximation. Thus 



o/r/L = fit^-Uy) +y 2 -2fiye- k y coskx 



+ !3 2 {i-k!,)e-' 2k y cos 2kx, . (53) 



