January 6, 19 16] 



NATURE 



509 



LETTERS TO THE EDITOR. 

 [The Editor does not hold himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

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 taken of anonymous communications.] 



Researches in Aeronautical Mathematics. 



The intimation in the columns of Nature that I 

 should be glad to receive offers of collaboration in the 

 solution ot problems in the applied mathematics of 

 aeroplanes has, as I am pleased to state, met with 

 a reply from Mr. Sely Brodetsky, lecturer in the 

 University of Bristol. 



Mr. Brodetsky has now definitely taken over the 

 problem of the two-dimensional motion of a lamina 

 in a vertical plane, with special reference to cases in 

 which the equations of motion can be integrated either 

 by methods of approximation or otherwise. The only 

 cases previously studied appear to be Lanchester's 

 "fugoid" and allied types of motion, and the small 

 oscillations about a steady state of motion in which 

 the plane of the lamina makes a small angle with the 

 direction of flight. Other possible types of motion 

 are those in which the lamina turns over and over in 

 its descent, or performs oscillations about a state of 

 steady descent in a vertical line with the plane of the 

 lamina horizontal. 



As these types of motion may occur when an aero- 

 plane becomes uncontrollable, it should be evident that 

 such investigations offer a considerable prospect of 

 leading to results of practical utility. 



Various empirical forms have been proposed for the 

 resultant pressure on a plane lamina or the co-ordinate 

 of its centre of pressure when expressed as functions 

 of the angle of attack. Among the former, Duche- 

 min's formula is probably the best known. It is con- 

 venient for purposes of comparison to choose the co- 

 efficients and constants so as to make the pressure 

 unitv when the angle of attack is 90°. Under these 

 conditions Duchemin's expression becomes 

 2 sin a 

 I + sin 'a 



The use of Fourier's series affords a convenient 

 method of standardising and comparing such expres- 

 sions, and the results of experiment. Duchemin's 

 formula is easily expanded by means of De Moivre's 

 theorem in the form : 



(4-2\/2)(sin a-fr sin 3a + r'sin 5a+ . . . .), 

 where r = 3 — 2^/2. 



Mr. T. G. Creak, of Llanberis, has evaluated the 

 coefficient in this series as well as in a number of 

 formulae proposed by other writers, and Mr. W. E. H. 

 Berwick has assisted. The investigation is now being 

 - ontinued by Mr. Caradog Williams, a post-graduate 

 student in my department, who is applying Fourier 

 series to the results of experiment, and, in particular, 

 to the tabulated results given by M. G. Eiffel in his 

 "Resistance of the Air and Aviation." 



The results, which are extremely interesting, indi- 

 'ate that the method is advantageous in several re- 

 -^pects; in particular, the Fourier expansions, while 

 being of uniform standard types, are sufficiently elastic 

 to be applicable to the most varied forms of plane and 

 curved surfaces. In the case of plane surfaces we 

 have the following results : — 



(i) The resultant pressure can be expanded as a 

 sum of sines of odd multiples of the angle of attack, 

 the expansion holding good from 0° to 360°. The 

 NO. 2410, VOL. 96] 



series usually converge according to the " inverse 

 square " law. The first coefficient usually lies between 

 10 and 125, and the second between o and 025. The 

 extreme limits, 1-25 and 0-25, represent the values 

 according to Soreau's formula, and are rather in 

 excess of those deduced from most of Eiffel's experi- 

 ments. 



(2) Por small angles of attack the sine series may 

 sometimes be inconvenient, and it is preferable to express 

 the result as the product of sin a into a series of 

 cosines of even multiples of a. From this cosine series 

 the series of odd sines is immediately deducible, but 

 the converse is not possible until the first (or con- 

 stant) term of the cosine series has been evaluated. 

 In future calculations it may, therefore, be better to 

 start with the cosine series. 



(3) The series for the lift and drift are immediately 

 deducible. 



(4) The distance of the centre of pressure from the 

 centre of area is given by a series of odd cosines, and 

 the moment of the resultant thrust about the centre 

 of area by a series of even sines. 



Mr. Williams and I are also employing the method 

 of least squares to obtain expansions in the case of 

 curved surfaces when the results are only required 

 for a limited range of tabulated values, and as the 

 method of least squares leads to Fourier's expansion 

 as a particular case, whatever be the number of terms 

 that it is desired to retain, comparison of results 

 should be easy. 



While it may be sufficient for many purposes to 

 adopt a formula of the type Aj sin a + A, sin 3a for the 

 resultant thrust, the first two terms are not sufficient 

 to give the correct position of the maximum in Eiffel 

 and Denes 's results for square plates. In very few 

 cases is there any advantage in going beyond sin 5a. 



The method is obviously applicable to an aeroplane 

 considered as a whole, and it thus opens up a number 

 of problems on longitudinal motion, equilibrium, and 

 stability. In particular the conditions may be studied 

 under which there is only one unique possible state of 

 steady motion. This condition does not lead neces- 

 sarily to the evolution of a non-capsizable aeroplane 

 analogous to the non-capsizable lifeboat, as equilibrium 

 can still be broken by the types of looping motion 

 mentioned earlier. 



The programme of work does not stop anywhere 

 near this point. The most important task before us 

 is to apply the method of initial motions to investigate 

 the effect of atmospheric disturbances — in other words, 

 suddent gusts of wind. For this problem the Fourier 

 method will be very useful in dealing with longitudinal 

 disturbances, but it is not so easy to decide on a suit- 

 able expansion when lateral changes of wind velocity 

 are taken into account. 



The motion of a kite was investigated by Prof. Bose 

 in the Bulletin of the Calcutta Mathematical Society, 

 ii., I ; unfortunately, however, an examination of the 

 paper by Mr. W. E. H. Berwick and myself reveals 

 a number of errors in the equations, and the form in 

 which the tension of the string is taken into account 

 renders the solution totaJly inapplicable to any system 

 resembling an ordinary kite. The only thing possible 

 in the circumstances was again to formulate the 

 correct equations of motion, which appear to be rather 

 complicated, entirely de novo, in the hone that Prof. 

 Bose or some other mathematician may be able to do 

 the rest. It would be an advantage if two workers 

 could attack the problem independently. It is not 

 encouraging to find that a problem which was sup- 

 posed to be relegated to the shelf as having been solved 

 requires to be reinvestigated. 



The effects of propellers on the equilibrium and 

 stability of aeroplanes require careful classification. 



