March 9, 1922] 



NATURE 



297 



raiding elegance of mathematical style and form 

 It are calculated to produce chaos and confusion 

 the study of stability for years to come. 

 In presenting students of aeronautics with the 

 5t treatise that is thoroughly imbued with the ideals 

 spirit of the best mathematical school of Cam- 

 idge, Dr. Brodetsky is performing for aviation what 

 erk Maxwell accomplished in anticipation of modern 

 trical engineering. In both cases the authors 

 ive undoubtedly been striking out on new lines and 

 iking fresh ground with scanty information to 

 ride them as to choice of subject-matter, method, 

 id order of treatment. The subject has thus opened 

 an almost illimitable collection of unsolved problems 

 which Dr. Brodetsky's treatise breathes in nearly 



line. 



As in electrical text-books, Dr. Brodetsky starts 



rith the simplest problems, but unlike them, appears 



avoid their subsequent contradiction of previous 



ilts. After a short introduction involving a 



lary of the problems presented by the heavier- 



i-air machine, section i, dealing with " Motion 



Air," opens with a chapter on " Dimensions," 



iling with elementary dynamical principles, in which 



count is taken of such possible influences as viscosity. 



this early stage important innovations in the matter 



notation become necessary. If the object of the 



iters who are continually changing their axes is to 



avoid copying the original stability monograph, they 



would have caused less confusion if .they had adopted 



ifferent letters. Now that Lr 



M„ or 



-Nr, it has become necessary to scrap the original 

 notation and substitute an entirely different set of 

 symbols. It is to be hoped that the same mistake 

 will not again be made, but that Dr. Brodetsky's 

 notation for the so-called " derivatives " and similar 

 quantities will be accepted as a permanent solution 

 for the present chaos. It certainly represents the best 

 that could be evolved after many hours of considera- 

 tion, in which the present reviewer took part when 

 studying with Dr. Brodetsky through a grant from the 

 Research Department under the hospitable roof of 

 Bristol University. 



The next chapter deals with problems on resisted 

 motion falling within the range of particle dynamics, 

 including a study of Lanchester's phugoid curves. 

 The reference to " catastrophic instability " on p. 53 

 is important in view of the frequent Press notices of 

 aeroplanes which are stated to have turned over 

 suddenly and crashed to the ground. 



Two-dimensional rigid dynamics forms the subject 



of the next chapter, which embraces not only the 



ordinary theory of longitudinal stability (including 



a rigorous proof of Routh's discriminant condition), 



NO. 2732, VOL. 109] 



but also applications to the kite and parachute. 

 Further, there is an original investigation of the 

 motion under gravity of a lamina under an assumed 

 simplified law of resistance which is illustrated by 

 diagrams deduced from theory and compared with 

 experiment. The reference to periodic solutions should 

 receive serious consideration. It must not be forgotten 

 that an aeroplane may be stable for small deviations 

 from steady motion and may yet be capable under suit- 

 able initial conditions of acquiring a periodic motion 

 turning over and over in loops until it crashes to the 

 ground, while it may be impracticable to extricate 

 it from this state. The aeroplane which effected a 

 successful landing after its observer and pilot had 

 been shot dead did not necessarily possess immunity 

 from this danger, which might have occurred if the 

 machine had started under different initial conditions. 



The next chapter deals with applications of three- 

 dimensional rigid dynamics. Here the range of 

 solved problems is necessarily limited. It, however, 

 includes circular and spiral flights and lateral stability 

 of the aeroplane, the parachute and the kite. In 

 these sections Dr. Brodetsky's notation is a great 

 improvement on its predecessors. It has the great 

 advantage that, hke the notation in the original 

 treatment of stability, it makes the terms of the 

 biquadratic all positive. In the same way it is much 

 easier to write down the pressure equation in hydro- 

 dynamics with the old-fashioned velocity potential 

 instead of the new one. 



As regards section 2 (" Dynamics of Air ") there 

 is less to be said. The one direction in which 

 mathematicians have made a serious attempt to do 

 substantial work more or less connected with aero- 

 nautics (often less) has been in applying hydro- 

 dynamics — mainly two-dimensional hydrodynamics 

 of perfect incompressible fluids — to the study of 

 pressures on moving laminae. In this connection Dr. 

 Brodetsky gives a good general treatment of dis- 

 continuous motion. The book as a whole, however, 

 is calculated to emphasise the importance of rigid 

 dynamics as applied to aircraft in their entirety in 

 contrast to these popular hydrodynamical investigations 

 of flow of air round parts of their structure, which, as a 

 matter of fact, are largely affected by mutual inter- 

 ference. It is to wind channel experiments that we 

 must look for the determination of the quantities 

 required in theoretical developments. 



In section 3 Dr. Brodetsky returns to the original 

 methods of treatment, and we are glad to notice 

 his recognition of the concept of the ideal " narrow 

 planes gliding at small angles " as having made it 

 possible to initiate a formal study of stability in 

 anticipation of the modern wind tumult. In the 



M I 



