We present in this paper a straight-forward method of deducing integral transform pairs directly from second order differential equations. In (1) this subject has been very thoroughly treated and in view of this we believe that the method described here has its greatest justification in its simplicity.

In an earlier paper (Gallagher & Mercer 1962) the numerical results for the first eigenvalue of the problem of plane couette flow were given. The higher eigen-values are now examined and are found to be in agreement with those of Southwell & Chitty (1930), but to disagree with those of Grohne (1954).

The problem considered here is concerned with small disturbances of plane Couette flow. As is usual in such problems it is assumed that the disturbance velocities are sufficiently small to allow the Navier-Stokes equations to be linearized. There results a special case of the well-known Orr-Sommerfeld equation and this is solved by an exact method using a digital computer. The problem has previously been considered by several authors, mostly using approximate methods and their results have been compared where possible with those obtained here. It was possible to proceed to values of αR not in excess of 1000 (α being the wave-number of the disturbance and R the Reynolds number of the basic flow), and the results tend to confirm the belief that Couette flow is stable at all Reynolds numbers.

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity

1.1

in which D 2 = d 2/du 2. Furthermore, when g is a solution of y″ + λ 2 y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to

1.2

In this note we generalize these results to the real Euclidean space EN , our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of

1.3

to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.