Tripos Question

The examination for part II, schedule B, of the Mathematical Tripos of 1910 was held at the Senate House and began at nine o’clock on Thursday, June 2. Question 8 of part C of the paper consisted of a typical, but daunting, combi­nation of book work and problem solving.7 The question read as follows:

C8. Prove that in irrotationally moving liquid in a doubly connected region the circulation is the same for all reconcilable circuits and constant for all time.

A long elliptic cylinder is moving parallel to the major axis of its cross sec­tion with uniform velocity U through frictionless liquid of density p which is circulating irrotationally around the cylinder. Prove that a constraining force Kp U per unit length of the cylinder must be applied at right angles to the di­rection of motion, where K is the circulation round the cylinder.

To a modern reader, versed in aerodynamics, the expression Kp U would be identified as the fundamental law relating the circulation, density, and ve­locity to the lift on a wing. In modern aerodynamics it is called the Kutta – Joukowsky relation. If interpreted aerodynamically, the “elliptic cylinder” would be a mathematically simplified substitute for the cross section of a wing, and the “constraining force” would be the weight supported by the lift. It is doubtful, however, whether any of the Tripos candidates of 1910 would have thought in this way. The year 1910 was when Kutta and Joukowsky, inde­pendently, published their results in German journals, and it is unlikely that the news had reached Cambridge. Admittedly “aeroplanes,” that is, aircraft wings, had been the subject of Tripos questions in the past, but the reference had been to Rayleigh’s paper on an inclined plate in a discontinuous flow, not to his tennis ball paper.8 Thus question C8 was unlikely to have evoked aero­nautical associations. If candidates attributed any technological significance to the formula KpU, it would have referred to ballistics not aeronautics. Perhaps some of the candidates had read Rayleigh’s tennis ball paper and

Greenhill’s extension of the analysis. More probably, they were calling up in their memories the relevant pages of Lamb’s Hydrodynamics and relying on the hours of coaching and drill to ensure that their recall was accurate. A well – prepared candidate would have remembered Lamb’s treatment of the irrota – tional flow of a perfect fluid around a circular cylinder. In article 69 of both the 1895 and 1906 editions, Lamb laid out his version of the analysis originally developed in the papers of Rayleigh and Greenhill. Like these writers, Lamb was mainly concerned with trajectories, but his analysis would have given the candidates both the general idea behind the question and the derivation of a formula which included an expression identical to that of the constraining force mentioned in question C8.

After explaining how the circulation augments the speed of flow on one side of the cylinder and diminishes, or even reverses, it on the other, Lamb had gone on to calculate the forces. Rayleigh had approached the problem using the stream function, while Lamb used the velocity potential. To begin, Lamb wrote down the velocity potential ф for the flow, assuming the cylin­der moving at any angle. He then differentiated this expression with respect to time to give дф/dt. Next he derived a term for q, the velocity of the flow. These results were substituted in a general form of Bernoulli’s equation to get a value for the pressure, and the pressure was then integrated round the sur­face of the cylinder to yield the resultant force. Lamb’s treatment was more general than Rayleigh’s, although there was no mention, even informally, of the role of friction. The two components of the force on the circular cylinder came out as follows, first in the direction of motion: and then at right angles to the motion:

KpU – M ‘Udx dt

where К is the circulation, p is the density of the fluid, U is the relative veloc­ity of the fluid and cylinder, M’ = npa2 represents a mass of fluid equivalent in volume to the cylinder with radius a, while % is the angle that the direc­tion of motion of the cylinder makes with the x-axis. The term KpU can be seen on the left-hand side of the second formula. Where the conditions of steady motion were specified as they were in question C8, the derivatives dU/dt and dx/dt, giving the rate of change with time, will be zero. The only remaining force on the cylinder will then be KpU at right angles to the motion.

Recollection of this result would have helped the candidates, but it would not have given them all they needed. The examiners of the 1910 paper had added a further complication to ensure that the mere reproduction of text­book material would not suffice. Lamb’s derivation referred to a circular cyl­inder, but the examiners had specified an elliptical cylinder. This made the question more difficult and required the candidates to demonstrate a facility with elliptic coordinates and elliptic transformations. If they could make the necessary transformation, they would then be in a position, for the rest of the deduction, to follow the pattern of the simpler case given by Lamb for the circular cylinder. Candidates would then have found that all the extra complexity actually produced terms that cancelled out, or went to zero in the course of the integration, thus leaving them, in the case of steady motion, with the same resultant force of Kp U.

Senate House records for 1910 show that A. S. Ramsey of Magdalene and A. E. H. Love of St. John’s were the two examiners who would have had re­sponsibility for the hydrodynamics questions. The other examiners, A. Berry of King’s and G. H. Hardy of Trinity, would have dealt with the more “pure” topics.9 Three years later Ramsey wrote a textbook on hydrodynamics in which circulating flow around an elliptical cylinder featured prominently.10 After discussing the case of the circular cylinder, and working through some of the intermediate steps in reasoning that Lamb had omitted, Ramsey showed the reader exactly how to address the problem of the ellipse. The move to el­liptical coordinates was explained along with advice about the types of func­tion that would satisfy Laplace’s equation and hence describe a possible flow. Ramsey also included question C8 from the 1910 Tripos paper in the exercises at the end of his chapter, which was called “Special Problems of Irrotational Motion in Two Dimensions” (119).

The title of Ramsey’s chapter conveys the point that I want to make. It shows the assimilation of Rayleigh’s tennis ball paper to the theory of perfect fluid flow in two dimensions. Ramsey did not wholly bypass the role of fric­tion. At the end of his discussion of the circular cylinder case he said: “The transverse force depending on circulation constitutes the mathematical ex­planation of the swerve of a ball in golf, tennis, cricket or baseball, the circula­tion of the air being due through friction to the spin of the ball” (101). Fric­tion was therefore mentioned, but the student was told that the mathematical explanation was to be found in inviscid theory. Ramsey, like Rayleigh, knew that this “mathematical explanation” could not furnish an account of how a spinning ball created the circulation. The point was implicit in the Tripos question which had two parts. The first part asked the candidates to prove that, under the conditions of the question, the circulation is constant for all time, that is, Kelvin’s theorem. The question then called on the candidates to generalize Rayleigh’s tennis ball result within this taken-for-granted inviscid framework.

It is now easy to see how G. I. Taylor could decide that Lanchester’s work was unacceptable. Taylor would have found himself confronted with some­thing very familiar and having little relevance for his research on eddies and turbulence. He would have known all about theories of circulation based on irrotational, perfect fluids and would have known how little they had to say about physical reality. He would certainly have been familiar with the math­ematical expression Kp U. It was the sort of thing that Tripos students were expected to deduce as an exercise in mathematical manipulation. Everyone in Cambridge knew that the cyclic approach gave an expression for a force but took away the possibility of generating that force. No wonder Taylor dis­missed the theory of circulation as readily as he dismissed the theory of dis­continuity. Lanchester’s theory was not a new discovery; it was the stuff of old examination questions.