variable. Now let u, (z, A), ; = 1, 2, 3, 4 be a suitable fundamental system of solutions 

 of (3) of known asymptotic form. Then I determine, in [1], four functions v } (z, A) of 

 the form 



Vi = fcjGOttj, j = 1, ■ • ■ , 4, (4) 



where the k } (z) are chosen so that the v } — vj(z, A) have, in some region away 

 from z = 0, the same asymptotically leading term as the solution of ( 1 ) . The functions 

 kj(z) may fail to be regular at z — 0, which makes the subsequent arguments more 

 involved. The related equation used in [1] is then the easily constructed fourth order 

 linear differential equation that has these v f , j = 1, • • •, 4 as a fundamental system. 

 By contrast, Lin computes a different set of four functions, which we shall also 

 call Vj, by setting 



Vj= CifeA)^ + C 2 (z,X)wj' + C 3 (z,AK" + C 4 (z,X)V', j = 1, • • • , 4 (5) 



and requiring that in a region away from z = these v, have the same asymptotic 

 form as a fundamental system of (1). Since the C } (z, A) are permitted to depend on 

 A, Lin has much more freedom in his construction so that he can construct functions v 7 - 

 that agree with the solutions of (1) up to an arbitrary but finite order (A~"0- His 

 coefficients are of the form 



m 



C,(z,\) = %r vn (z)\-", (6) 



n=l 



and it turns out that the r Vn (z) can be chosen so as to be regular at z = 0. These are 

 important improvements. 



The work of part 3 of the general procedure outlined above, in which the 

 related and the given equations have to be compared as to their asymptotic form in 

 a domain including the turning point z = 0, has not yet been fully carried out for Lin's 

 method. A well developed technique for this task exists, but its application to Lin's 

 differential equations may still involve considerable labor. 



T. B. Benjamin 



I believe it may be appropriate here to refer to another problem of hydro- 

 dynamic stability, one which has received only little attention compared with the stability 

 problems for the boundary layer and pipe flow, which have been in the forefront of 

 theoretical hydrodynamics for about a half a century now. 



The case I would like to mention is the flow of a viscous liquid in a thin film, 

 bounded on one side by a plane wall, and on the other by a free surface. Flow of this 

 type is a fairly common experience. 



For instance, rain water may sometimes be observed running in a thin sheet on 

 the outside of a window. Again, one may often observe motor oil, or paint, or mayon- 

 naise, flowing in a thin sheet down the sides of some solid object immersed for sampling 

 purposes. 



The general mechanics, and in particular, the stability of such flows, is an im- 

 portant question in chemical engineering, for instance, particularly where the film is in 

 contact with a gas stream. 



Some work on this problem has been done recently in Cambridge, both in the 

 Chemical Engineering Department and in the hydraulics laboratory, and I would like 

 to outline what the problem was, briefly, and what progress has been made on the 

 theoretical side at least, since this seems the aspect most appropriate to this session of 

 the Symposium. 



It is a well-established fact that when a film of water runs down a vertical plate, 

 the flow is laminar and uniform until the Reynolds number, based on film thickness, is 

 raised to a value of the order of unity. Waves with a well defined periodicity then 



365 



