NATURE 



535 



THURSDAY, JANUARY i6, 191 3. 



.4 MATHEMATICIAN'S LECTURES ON 

 AERONAUTICS. 

 The Dynamics of Mechanical Flight. Lectures 

 delivered at the Imperial College of Science and 

 Technology, March, igio and 1911. By Sir G. 

 Greenhill. Pp. iii+121. (London: Constable 

 and Co., Ltd., 1912.) Price 8s. 6d. net. 

 T TP to the present time the study of problems 

 Lv relating to aeroplanes and airships has con- 

 spicuously failed to atti'act the attention of our 

 leading mathematicians and mathematical phy- 

 sicists. This is the more surprising in view of the 

 important part that has been played in the past, 

 and is still being played, by methods of mathe- 

 matical analysis in systematising and elucidating 

 our knowledge of electric phenomena. A book by 

 so trustworthy a mathematician as Sir 0. Green- 

 hill should prove of great value in clearing up the 

 misunderstandings which have so frequently arisen 

 as to the meaning and use (or misuse) of formula; 

 in connection with aeronautics. 



This book claims to be the substance of the 

 lectures given by Sir G. Greenhill at the Imperial 

 College, and in view of the difficulties of writing 

 a book of this character, the author has probably 

 valid reasons for not wishing to extend its scope 

 beyond the limits of these lectures. What he has 

 done is to present in the first place a simple account 

 of some of the more elementary problems which 

 are discussed in detail in his report on the stream 

 lines past a plane barrier, and in the second place 

 a general summary of the formulas that are in- 

 volved in relations between lift, drift and horse- 

 power, gyrostatic action, the screw propeller, and 

 the pneumatical principles of the airship. As 

 might naturally be expected by those who know 

 Greenhill's writings, the introduction is mainly 

 taken up with quotations from Greek, English, 

 and other classics. 



There are two methods of investigating the pres- 

 sure on a plane moving through a fluid medium. 

 One is the Newtonian method, which expresses the 

 pressure in terms of the momentum communicated 

 to the column of air on which the plane impinges. 

 The other is based essentially on Bernouilli's pres- 

 sure-equation, which determines the pressure from 

 the stream line motion past the plane. 



The Newtonian method still finds great favour 

 with a large class of practical men, and many 

 attempts have been made to apply it to aeroplanes. 

 But beyond Newton's deduction of the so-called 

 " sine-squared " law for a medium which, as Green- 

 hill remarks on p. 14, is taken "to behave like a 

 cloud of particle dust," little or no progress has 



NO. 2255, VOL. go] 



been made in obtaining results that can be re- 

 garded as established on a trustworthy 

 basis. On p. 41 Greenhill gives a figure show- 

 ing a popular misrepresentation of the stream lines 

 past a cambered plane, which finds favour with 

 some of these neo-Newtonian would-be philoso- 

 phers, but in which the absence of any broad- 

 ening of the stream lines is inconsistent with the 

 existence of any thrust or lift. 



The theorv of discontinuous motion as originated 

 by Kirchhoff and Helmholtz, and developed by 

 Lord Rayleigh, Schwarz and Christoffel, Michell, 

 Love, and finally at very great detail in Greenhill's 

 report, comprises all the problems that are soluble 

 by existing analytical methods regarding fluid 

 pressures on planes, as determined from 

 Bernouilli's equation. It is true that they involve 

 assumptions which at best form only a first 

 approximation to the conditions prevailing in actual 

 aeroplanes. But" a closely analogous relation 

 exists between the application of conjugate func- 

 tions to electrical problems and the calculations 

 required in practical electric engineering. No one 

 will question the value of analytical applications 

 of complex variables to electricity, and similarly 

 these hydrodynamical investigations offer the best 

 basis for a theoretical explanation and study of 

 aeroplane pressures. 



A considerable portion of the chapters on this 

 subject is devoted to a discussion of the integrals 

 involved in problems where the first integration can 

 be eiTected without the use of elliptic functions. 

 The condition for this is that there must only be 

 two edges at which discontinuities are formed, 

 or that the number of such edges must be reducible 

 to two from considerations of symmetry. In such 

 cases the "Omega" diagram representing the 

 logarithm of the reciprocal of the velocity has 

 only two right-angles, and the first integration 

 only, involves a quadratic expression under the 

 radical sign. A table of different forms of this 

 integral is. given in the present book. Since the 

 book appeared, some researches have been started 

 at Bangor on the lift and drift of bent planes, with 

 the view of estimating the effects of camber. One 

 such application is suggested by Greenhill on 

 p. 40, who, however, assumes that the stream 

 lines divide at the bend; this we find is not the 

 general rule, but a particular case. The object 

 of the research is to obtain numerical tables of the 

 lift and drift in particular systems, and so to 

 acquire some information regarding their relative 

 efficiency. The integrals are all capable of evalua- 

 tion, and this research could probably not have 

 been started but for Greenhill's previous work on 

 the subject. 



The chapter on gyrostatic action is of an ele- 



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