370 Lord Kayleigh on the 



i ^t *= - 2k/3 2 + 2y -2/3 (1 -%>"** cos ib 



-2£/3 2 (l-%)*- 2 ^cos2/b;, (54) 



solutions applicable also to the problem of the elastic plate, if 

 'xjr be understood to mean the transverse displacement. 



In the above investigation, so far as it applies to the hydro- 

 dynamical question, L 2 has been supposed to be negligible. 

 We will now retain the square of L, but simplify the problem 

 in another direction by neglecting the square of j3, so that the 

 first approximation is 



f=Lf-2Lj3ye- k ycosk%. . . . (55) 



The exact equation (derivable from (34)) for the motion of 

 a viscous fluid in two dimensions is 



vV = H^ + »f!p (56) 



T v doc v ay ' 



From (55), 



V 2 -f = 2L + 4L&/3e-^cos&.z, 



^ JVV + ,^° = _8L 2 P^r^ sin h'. . . (57) 

 (xx ay 



Using this in (56) we have 



v ^ = _^!^ 2 ^-% s in^. . . . (58) 

 The solution of 



($-*?h =^~* (59) 



is 



t^_ £<r* (60) 



r $k 3 + 24F ' v ' 



so that the required solution of (58), correct as far as the term 

 involving L 2 , is 



^ = Ly 2 -2L ) ey6-^cos^-^— (^ + Jfy 3 )«-*ysin&*. (61) 



It may be well to repeat that, though L 2 is retained, /3 2 is 

 neglected in (61) ; that is, the depth of the corrugations is 

 supposed to be infinitely small. 



The part of the motion proportional to L 2 is, of course, 

 independent of the direction of the principal motion of the 

 fluid, and is thus in a manner applicable even when the 

 principal motion is alternating. With regard to the relative 



