NA TURE 



[April 22, 1909 



direction of motion and proportional to the square 

 of the velocity. 



In view of the fact that classes on calculus for 

 eng-ineers form an integral part of every modern 

 technical course of instruction, it is to be regretted 

 that Mr. Lanchester has not made some attempt to 

 bring his equations more into conformity witli ordinary 

 well-recognised notation. When he writes down the 

 equation of his curve as 



and the expression for the radius of curvature as 



rfB' 

 his readers will take some time to find out what these 

 equations mean ; whereas any student who has 

 attended the classes above referred to would under- 

 stand at a glance the same equations if written as 



— = -=!- +— and p= ,-■ 

 ds 31-0 sly d<p 



When Mr. Lanchester applies his phugoid theory 

 to investigate the longitudinal stability of aerofoils, 

 he at once comes into conflict with the theory which 

 the present writer, in conjunction with Mr. VVilliams, 

 worked out some few years ago. There has been 

 some difficultv in making out how Mr. Lanchester 

 arrives at his results, and Mr. Harper has examined 

 the matter independently. It was not intended at first 

 to deal with, what might be a controversial point in a 

 review in Nature, but the difference between the two 

 methods is probably not so very difficult to explain. 



According to our theory, the small oscillations about 

 steady motion of an aeroplane, or indeed any body 

 moving in a resisting medium in a vertical plane, 

 depend on the roots of a biquadratic equation, and 

 the conditions for stability are those given by Routh. 

 This method enables account to be taken of every 

 circumstance which may affect the stability, in parti- 

 cular, variations in the position of the centre of 

 pressure for different angles. 



Mr. Lanchester, on the other hand, considers only 

 the case of a single aeroplane, the variations of 

 the centre of pressure of which are not taken into ac- 

 count, stability being maintained by means of a tail. 

 He starts with the assumption that his "phugoid" 

 oscillations, when small, are simple harmonic in 

 character, and that the effect of the moment of inertia 

 of the machine, as well as of resistances, is to change 

 the amplitude of these oscillations. In estimating 

 the effect of these changes he assumes the equations 

 of simple harmonic motion to hold good. For 

 example, in considering the rotatory motion about 

 the centre of gravity (§ 63) he writes 



T, = ?I^%20, = 40iI^', 



where Tj is the maximum torque, Oj the maximum 

 angular displacement, fj the time of oscillation (cal- 

 culated, it would seem, from the ideal " phugoid " 

 motion), and I the moment of inertia. 



This step is unjustifiable. The correct equation is 

 . d-e 

 dt^ 

 where r is the torque at any instant. A similar 

 consideration applies higher up, and when the neces- 

 sary corrections have been made they are found to 

 lead to the biquadratic equation of the Bryan- 

 Williams theory. 



It is as if, in working out the theory of the com- 

 pound pendulum, it were attempted to treat one 

 weight as a simple pendulum, and to assume that 



NO. 2060, VOL. 80] 



the motion of the other weight did not affect the 

 period, or the relation between velocity and displace- 

 ment, but merely produced variations of amplitude. 



Mr. Harper has applied the Bryan-Williams method 

 to the particular kind of tailed aeroplane considered 

 by Mr. Lanchester, and he obtains a numerically 

 different result, the discrepancy being accountable for 

 by the assumptions contained in Mr. Lanchester's 

 method. 



There is thus a good bit of work of a theoretical 

 character requiring to be done before the problem of 

 stability can be regarded as completely solved. In 

 the meantime, it must be remembered that airship 

 designers have not, as a rule, undergone even the 

 elementary training in practical mathematics referred 

 to in this review, and that most extraordinary views 

 commonly prevail in this country regarding the sub- 

 ject of stability. It is not improbable that Mr. 

 Lanchester's conditions may be sufficiently near the 

 mark for practical purposes, and his experimental 

 verifications seem to support this view. Moreover, 

 they may err on the side of safety. It seems also 

 certain that unstable machines have been safely guided 

 through the air by skilled manipulators, and the 

 stability of the Wright machine has been seriously 

 questioned. Indeed, there are good reasons for be- 

 lieving it to be unstable. Mr. Lanchester's method 

 applied to the Lilienthal machine shows it to be 

 unstable, although, in view of its broad curved sup- 

 porting surfaces, a complete investigation would 

 require account to be taken of several neglected 

 factors for which no experimental data exist. 



A great deal of rubbish has often been written on 

 the " soaring bird," and much that has been stated in 

 print has been incompatible with the doctrine that 

 perpetual motion is impossible. Mr. Lanchester's 

 observations and experiments are deserving of the 

 most careful consideration, and the same applies to 

 his chapter on " Experimental Aerodonetics." The 

 book represents the result of a serious effort to place 

 the theory of flight on a scientific basis, and should 

 do much to convince would-be aviators that " airship 

 design " can no longer be regarded, as it has been in 

 the past, as a mere exercise for the imaginative 

 faculty, but as a subject requiring' hard thought, 

 endless experiments, and great care in drawing con- 

 clusions from them. 



Sir Hiram Maxim's book is distinctly disappoint- 

 ing. An account of his early experiments, if some- 

 what out of date, would be at least of historic in- 

 terest ; but when the author indulges in a tirade 

 against mathematicians, the question which naturally 

 suggests _itself is, Where on earth does he find his 

 mathematicians? He tells us that 



" During the last few years a considerable number 

 of text-books have been published. These have, for 

 the most part, been prepared by professional mathe- 

 maticians who have led themselves to believe that all 

 problems connected with mundane life are susceptible 

 of solution by the use of mathematical formulae, 

 provided, of course, that the number of characters 

 employed are numerous enough." 



Now Prof. Chatley, who certainly has got a pretty 

 clear grasp of the present state of the flight problem, 

 recently wrote : — 



" A few great mathematicians (including Lords 

 Kelvin and Rayleigh) have devoted some attention to 

 the matter, but the author is not aware that any 

 mathematician worthy of the name has considered it 

 worth while to make an exhaustive study of the 

 question " 



When, Sir Hiram says that " Up to twenty 

 years ago Newton's erroneous law as relates to 

 atmospheric resistance was implicitly relied upon. 



