224 SCIENCE PROGRESS 



f r (a), <j) r (a) can be found experimentally. For more than one 

 plane we add the separate effects. 



Theory to Find f(a) 



Fix CP = a<£(a) = o, i.e. take the centre of pressure as the 

 arbitrary point. Duchmein gives 



R=2R 9°'7+ 1 sin'a = 2l V sina (approximately). 



Prof. Bryan then obtains f(a) <x R, f(a) = sina. 



H = X = SKSlPsin'a, - Hh = N Q - U 2 2KS£'sina, 

 W = Y = SKSU 2 cosasina, 



where l- = xcosa — ysina, f" = xcosa — 2ysina for brevity. 



The sign 5" refers to more than one plane. 



If the planes are narrow — as assumed above — f r (a), a<£ r (a), and 

 a(f>'(a) are negligible. 



Allowances as to the above values must be made for the 

 inclination (77) of the thrust H with the axis Gx, for head resist- 

 ances, for propeller effects, and for the effect of the direction of 

 flight relative to the horizon on the derivatives. See Prof. 

 Bryan's work (pp. 75-122). 



Development of the Asymmetrical Derivatives 



See Prof. Bryan's work (pp. 123-164). 



The law used is that of Newton. Resultant pressure on 

 element dSoc resultant velocity x normal velocity of air relative 

 to the machine. 



An element dS is taken at (xyz) so that the direction cosines 

 of the normal to it are 1, m, and n. The velocity of the plane is 

 U when increments u, v, w, p, q, and r are added. 



X =/Kl 2 U 2 dS + 2Uu/Kl 2 dS + Uv/lmKdS + Ur/l(mx - 2ly)KdS 



Y = IP/KmldS + 2Uu/KlmdS + Uv/m 2 KdS + Ur/m(mx - 2ly)KdS 



Z = Uw/n 2 KdS + Up/Kn(ny - mz)dS + Uq/n(2lz - nx)KdS 



L = U\v/Kn(ny - mz)dS + Up/(ny - mz) 2 KdS + Uq/(ny - mz)(2lz - nx)KdS 



M = Uw/Kn(lz - nx)dS + Up/K(ny - mz)(lz - nx)dS 



+ Uq/K(2lz - nx)(lz - nx)ds 

 N = U VKl(mx - ly)dS + 2Uu/Kl(mx - ly)dS + Uv/m(mx - ly)KdS 



Ur/(mx - ly)(mx - 2ly)KdS. 



