February 4, 1909] 



NATURE 



415 



By substitution between (7) and (8) the speed corre- 

 sponding to a given power and angle may be obtained, or 

 the power required to drive the machine at any particular 

 angle and speed. If the machine be rising, so that the 



Ki^ntue Ti"ti,sT - OivrcT/pirrifniK-^ 



Flc. 3.— Equilibrium of Forces in Aeroplanes: Aeroplane Descending 

 (iliding). 



line of motion is inclined at an angle S to the horizon, 

 then (8) becomes 



W = Pccs(9 + 7) = 2/!-SV-sin 7C0S (6 + 7 . . (9) 

 Bv substitution between (7) and (9) we can find the power, 



Fic. 4. — Trajector 



speed, and angle in terms of one another in the new 

 ■circumstances, which arc the most adverse that have to 

 be considered. 



Power Required for a Helicoptere. 

 This will follow at once from a consideration of pro- 

 peller thrust. For if T be the thrust in lb. of a propeller, 

 under given conditions as to speed and slip, then in a 

 helicoptere 



W = «T (10) 



where 12 is the number of propellers. 



The ornithoptere will be discussed later. 



Efficiency of Propellers. 

 E.vperiment has generally shown that, subject to correc- 

 tion for the difference of density, an air propeller is almost 

 identical in its action with a marine propeller. The thrust 

 is proportional to the area of the blades and the square 

 of the speed, and the power varies as the cube of the 

 speed. There is a diminution of thrust with a decrease of 

 slip, and both power and thrust increase with the diameter 

 of the propeller. There is no necessity to present here the 

 general conclusions as to propellers, which will be found 

 in Mr. Froude's papers in the Trans. Inst. Naval .Archi- 

 tects, and in te.xt-books on naval architecture and marine 

 engineering. There is, however, one respect in which the 

 action of a propeller in air differs from that in water, viz. 

 the feed. Owing to the small inertia of air, a propeller, 

 revolving on a fixed axis in air previously stationary, 

 rapidly ejects air by axial propulsion and centrifugal force, 

 and tends to surround itself by a vortex of air, with a 

 consequent diminution of the thrust to almost zero. This 

 is the reason for the lack of success in experiments which 

 liave been made on liftnig screws for hclicopt^res. On 

 I he other hand, an axial or transverse flow caused by 

 inolion of the axis of rotation will supply the propellers 

 with ihc necessary fresh air, and consequently we find 

 dial tlie smaller the slip (i.e. Ihc greater the advance) of 



NO. 2040. VOL. 79] 



the screw the greater its efficiency. Similarly, in heli- 

 copt^res moving laterally there is more lift. 



For a sustaining screw not rising (i.e. with 100 per 

 cent, slip) the author has deduced the following formula 

 for the thrust (see " The Problem of Flight," p. g) : — 



;hd" 



(II) 



where T is the thrust in lb., r the revolutions per second, 

 H the horse-power, and D the diameter in feet of the pro- 

 peller. This is based on the assumption that the area is 

 that required by the conditions as to power, diameter, and 

 speed. The following rule for the projected area must be 

 applied : — 



4 2 IT 



vS <'^) 



A = — 



where A is the ratio of the projected area to the disc area. 



These rules are based on Mr. W. G. Walker's experi- 

 ments with fans, particulars of which will be found in 

 .Mr. Innes's book on " The Fan." 



The thrust per horse-power obtained with the best forms 

 of propellers varies from 20 lb. to Go lb., 40 lb. being the 

 common ma.ximum. The mechanical efficiency, as in the 

 case of marine propellers, rarely rises above 50 per cent., 

 the best results being obtained with a minimum of slip. 

 This alone gives the aeroplane a superiority over the 

 helicoptfere.' 



Stability of Gliders. 



We have seen that the centre of pressure is ahead of the 

 centre of area, and that the distance between these two 

 depends on the angle 7. If then the angle and the normal 

 pressure are constant, the turning moment of the pressure 

 about the centre of area is also constant, and may be 

 balanced by shifting the centre of gravity until it lies over 

 the centre of pressure. Seeing, however, that neither the 

 angle nor the resistance is absolutely constant, it might 

 be supposed that stability was impossible. That this is 

 not so has been demonstrated by Prof. Bryan and Mr. 

 Williams in a paper read before the Royal Society in 1903, 

 and by Captain Ferber in an article in the Revue 

 d'.irtillerie (November, 1905). In the latter it is shown 

 that an aeroplane is longitudinally stable if two conditions 

 are satisfied. 



(i) That the longitudinal radius of gyration about an 

 axis through the centre of gravity does not exceed 



v 



iT' 



(13) 



when P is the weight of the aeroplane in kilograms, and 

 b the overall width of the machine in metres. The radius 

 of gyration is here measured in metres. 



(2) That the centre of gravity falls over the centre line 

 between two points, one a little ahead of the centre of 

 area of the sustaining surfaces, the other near the for- 

 ward edge of the aeroplane. The exact values of these 

 positions depend on the characteristic magnitudes of the 

 machine through a series of somewhat complex equations, 

 for which the papers referred to should be consulted. It 

 must be recognised in this connection that the probable 

 inaccuracy of Joessel's formula invalidates the accuracy 

 (110/ the method) of the values given by Captain Ferber 

 in this paper. 



If the centre of gravity coincide with one of these points, 

 the machine is subject to two oscillations of long and 

 short periods respectively, any increase of which will lead 

 to collapse. 



The behaviour of a machine running with a certain 

 initial speed is then somewhat as follows. The continued 

 resistance tends to retard the machine, and to cause the 

 velocity to fall below the soaring limit, and the weight 

 (in front of the centre of area) causes the front to dip. 

 The gravitationally acquired velocity causes a forward 

 acting pressure on the surface, so that if the machine is 

 stable (in accordance with the above conditions), it settles 

 down into a condition in which the resistance due to the 

 resultant velocity just balances the component of the weight 

 in the direction of motion. P^naud has shown that the 

 angle between the plane and the direction of motion 

 (trajectory) {I'atiglc d'altaqiic) is half the angle between 

 1 See the author's paper to the .Aeronautical Society, October, 1908. 



