whose asymptotic theory is already known and whose solutions are conjectured to have 

 the same asymptotic form as those of the given equation, at least up to terms of some 

 given order 0(A~'")- Since the asymptotic character of the solution of such differential 

 equations is apt to change abruptly from one region of the complex plane to another 

 (so-called "Stokes phenomenon"), it is often difficult to find a related equation whose 

 solutions resemble asymptotically those of the given equation in a domain as large as 

 the physical applications of the theory require. Only very simple types of differential 

 equations can be directly analyzed asymptotically. The most common tools for this 

 are integral representations resulting from functional transformation such as the Laplace 

 transformation. Therefore, the asymptotic investigations along the lines described here 

 begin frequently with the study of some special differential equation L s [v, A] — for 

 which a direct asymptotic theory can be given. 



The complete procedure consists then of the following parts. 



1 . Choice and asymptotic analysis of a suitable special differential equation L s [y, A] = 0. 



2. Construction of a suitable related differential equation L r [y, A] = based on 

 L s [y, A] = 0. The related equation should be such that the asymptotic form of its 

 solutions follows readily from that of L s [y, A] = 0, e.g., by a transformation of 

 variables. 



3. Comparison of L, [y, A] = and the given differential equation L g [y, A] = 0, i.e., 

 proof that the solutions of the two equations differ by terms that are 0(A~'")> m ^ 0, 

 in the considered domain of the independent variable z- This part frequently re- 

 quired tedious appraisals. 



The given differential equation studied by Lin is of the form 



L g [y,\] = y w +\* [P(z,X)y"+QXz,\) y'+R (z,\) y]=0. (1) 



This is somewhat — but not essentially — different from the differential equation treated 

 in [1]. The Sommerfeld-Orr equation is a special case of (1). In order to obtain the 

 results needed in the hydrodynamic applications, z should be permitted to range over 

 a domain containing one simple zero of P(z, go). Let us assume, e.g., that 



P(o,«0=0, f— J ^0. (2) 



V dz / s=0 



A=oo 



It is the inclusion of this "turning point" z = into the z-domain that causes most of 

 the difficulties. 



The simplest differential equation of the type (1), (2) which still has all its 

 essential features is 



L s [y,M = y w +\Ky"+ay'+0y), a,p constants. (3) 



This is the special equation used by Lin and — with a = — by myself. Its full asymp- 

 totic theory, for a = 0, was developed by myself in [1] and in [2], Annals of Math, 

 v. 52, pp. 350-361 (1950), by means of complex Laplace transformation. 



In a sufficiently small region not containing the turning point asymptotic expan- 

 sions for the solutions of (1), and hence of (3), have been known for a long time (cf. 

 [3], Annals of Math, v. 53, pp. 852-871 (1948)). Both Lin and myself use these 

 expansions for the construction of a related equation. First, a preliminary transforma- 

 tion of the independent variable has to be performed. In order to explain the meaning 

 of this transformation without entering into technicalities we refer to the fact, proved, 

 e.g. in [3], that the previously mentioned abrupt change of asymptotic character takes 

 place upon the crossing of certain curves ("transition lines"), which, in the case of the 

 equation (3), form three rays issuing at equal angles from z = 0. By properly trans- 

 forming the independent variable the transition lines of ( 1 ) can be made to coincide 

 with those of (3). Let us assume that (1) is already written in terms of this new 



364 



