Greenhill’s Memorandum

Greenhill’s task was to carry Rayleigh’s mathematical analysis forward. The result was the Advisory Committee’s lengthy Reports and Memoranda No. 19, published in 1910. Its full title was “Report on the Theory of a Stream Line Past a Plane Barrier and of the Discontinuity Arising at the Edge, with an Application of the Theory to an Aeroplane.”8 (The word “aeroplane” here means “wing.”) Greenhill addressed neither the empirical shortcomings of Rayleigh’s model (acknowledged by Rayleigh himself), nor the issue of instability and turbulence in the flow that had been raised by Kelvin (and brushed aside by Rayleigh). He discharged his duty by assembling everything that was known about the mathematics of discontinuous flow. As Greenhill put it, “the object of the present report is to make a collection of all such problems solved so far, and to introduce a further simplification into the treatment” (3). It was not unsolved problems but solved problems and their further refinement that engaged Greenhill’s attention. Particular attention was given in the report to the work of two other Cambridge mathemati­cians, Michell and Love. J. H. Michell was fourth wrangler in 1887. He be­came a fellow of Trinity in 1890 and a fellow of the Royal Society in 1902. In 1890 Michell had written a seminal paper titled “On the Theory of Free Stream Lines.”9 A. E. H. Love’s Cambridge credentials have been mentioned in chapter 1. His paper “On the Theory of Discontinuous Fluid Motions in Two Dimensions” was published in 1891 and provided a development of Michell’s work.10

Michell and Love introduced two new methods into the repertoire for turning free streamlines into simple straight line flow. One method was to make a transformation by taking logarithms. Such a transformation has the effect of turning the arc of a circle into a straight line. It led to a new dia­gram of the flow which had the angle of flow as one axis and the logarithm of the reciprocal of the speed as its other axis. In this way the streamlines were turned into a polygon in the extended, mathematical sense of the word. The other contribution was to make explicit use of the Schwarz-Christoffel theorem, which put the process of finding the necessary transformations on a more systematic basis. Compared with Rayleigh’s original discussion, the

number of transformations had been increased, but the new approach gave the analysis a more routine character, and Greenhill (see fig. 3.3) applied it, indefatigably, to case after case.

There is no doubting the mathematical sophistication of the material that Greenhill gathered together. The report was a virtuoso display of wrangler skills. One could say that it was Tripos aeronautics in full flight—but for one oddity. Where were the airplanes? The cases he collected appear to have little to do with aeronautics. Apart from comparisons with electrical phenomena, drawing heavily on the work of Maxwell and J. J. Thomson, the bulk of the examples treated themes such as the flow of water through orifices, spouts, and mouthpieces. Jets of water impinge on plates, and water flows through channels, around barriers, past piers, and over weirs. Walls, bridges, and pil­lars feature more prominently than flying machines. The puzzle is not the extent to which the discussion deals with water rather than air; these can properly be dealt with together. The problem is why this particular range of examples has been introduced. Given that Greenhill had little to offer that might directly strengthen the connection between Rayleigh’s mathematics and the wings of aircraft, what did he take himself to be doing?

The clue lies in the diagrams. All the cases that Greenhill discussed could be reduced to simple configurations of straight lines. They were all shapes that could be turned into the “polygons” needed for the application of the Schwarz-Christoffel theorem. Interpreting them as “mouthpieces,” “reser­voirs,” “weirs,” “piers,” and the like was distinctly post hoc. This was espe­cially true in the few cases where an attempt was made to link the diagram to aeronautics. Thus Greenhill gave an analysis, analogous to Rayleigh’s, of a flow against an inclined plane, but the main line in the diagram was posi­tioned between two further lines, one above it and one below it. This, said Greenhill, “may be taken to represent a rudder boxed in” between two planes (17). A variant of this figure effectively dispensed with the upper line by lo­cating it at infinity, “so that the analysis will serve for an aeroplane flying horizontally near the ground” (20). The words “may be taken to represent” and “will serve for” reveal the derivative character of the interpretation. The “indirect method” was at work. The examples had been gathered, not because of their relation to wings or aircraft, but because of their relation to a cer­tain, favored mathematical technique. Relevance to what Greenhill called the “analytical method” of the report—which from the outset he identified as the deployment of the Schwarz-Christoffel theorem—was the real principle of selection.11

Greenhill’s report to the Advisory Committee was modeled on a species of document characteristic of Cambridge mathematics and perhaps unique to it. It had become customary for the examiners of the Mathematical Tripos to publish the questions they had set in the previous year, compiling books of problems along with their approved solutions. This practice contributed to a cumulative archive of mathematical work which progressively deepened and refined the Tripos tradition.12 The archive was vital for the coaches in honing the skills of the next cohort of would-be wranglers. It enabled them to identify the main theorems that would be tested so that they could teach their students to recognize all the possible applications of the result, however diverse the fields, and disguised in outer form, they might be. Routh had published such a collection in i860 when he had acted not only as a coach but also as an examiner.13 The most famous collection was Joseph Wolsten – holme’s Mathematical Problems of 1867.14 Greenhill himself, after his stint as a Tripos examiner, had published Solutions of the Cambridge Senate-House Problems and Riders for the Year 1875.15 The 1910 report was just such a col­lection of problems and solutions. One hears the voice of the conscientious coach as Greenhill provided useful hints to his readers to help them avoid errors and traps. “The signs are changed when the area is to the left hand,” warned Greenhill, “so it is useful to employ an independent check of the sign” (5). “It simplifies the work to take i = <x>” (6). Again, “we introduce an angle ф, not to be confused with ф the velocity function” (10).

Although its contribution to aeronautics was close to zero, Greenhill’s R&M 19 soon joined the papers of Michell and Love in the list of canonical sources that were cited in Horace Lamb’s Hydrodynamics.16 A. S. Ramsey, of Magdalene College, who had been seventh wrangler in 1889, did not mention Greenhill by name but introduced his extensive discussion of discontinuous flow in his 1913 Treatise on Hydromechanics by saying, “such problems have recently acquired a new interest because of their relation to Aerodynamics.”17 Others of a more practical bent were less appreciative. The review in the Aeronautical Journal for 1911 was signed “B. G.C.”—presumably Bertram G. Cooper, who was to become the editor in 1913. Cooper was exasperated by Greenhill’s report: there were 96 large-format (“foolscap”) pages of text and 13 sheets of diagrams with, “on the average, about 8 lines of the vernacular to each fsc. page, the rest being mathematical equations.”18

It would doubtless be expecting too much of human nature to ask that the mathematician and the practical man should make up their minds to co­operate. Only, however, by a reasonable combination of the methods of both can the best results be obtained. If, therefore, the Advisory Committee were to lay their heads together and produce a volume giving a quantitative compari­son between solutions of problems as calculated mathematically and as obtained by actual experiment, they would clear the ground enormously, and inciden­tally would do something towards fulfilling the function which the average man (doubtless from the depths of his ignorance) considers that they exist to perform. The publication of an expensive work, such as this, giving no results or deductions in English, is highly to be regretted. (94)

The anonymous reviewer in Flight was equally aghast, calling it the “most extraordinary book yet published relating to the subject of aeronautics.”19 It would be unintelligible to 9,999 out of every 10,000 potential readers. Would some other member of the Advisory Committee, asked the reviewer, please write a nonmathematical report explaining the “practical deductions” to be drawn from Greenhill’s work? The reviewers in the technical journals clearly believed that the Advisory Committee was throwing down the gauntlet to the practical men, and their reaction was predictable. It would seem, however, that these robust responses from the nonmathematical reviewers had an ef­fect. Subsequent publications, when not entirely empirical, typically involved a comparison of theory and experiment. Nothing quite like Greenhill’s report was seen again.20