THE DISTURBED MOTION OF AN AEROPLANE 225 



Here we see that (XYN) uvr are the symmetrical group and 

 (ZLM) wpq are the asymmetrical group, when the plane is sym- 

 metrical to z = o, and when D = O = E, odd powers of z and n 

 being neglected in the above. 



For planes bent up at an angle ft the direction cosines of the 

 normal are sina, cosacos/3, cosasin/3. Where ft = o — i.e. the 

 planes are normal to the plane z = o — these reduce to sina, 

 cosa, o, and we see that X u , Y u , etc., reduce to the values already 

 found (p. 224). 



Asymmetrical Derivatives (ft = o) 



Let I = the moment of inertia of the plane with respect to xy 

 similarly (density = 1), then I =/z 2 dS. 



L p = KUIcos 2 a, L = - 2KUIcosa sina, 



M p = -KU I sina cosa, M q - 2KUIsin 2 a. 



Prof. Bryan concludes that fins are needed for stability in 

 this case. 



A Single Fin 



Let its area = T; K' its coefficient of resistance ; Xi, y l( z x the 

 co-ordinates of its centre of pressure (1 = o, m = o, n = 1 — i.e. 

 it is parallel to the plane z = o). 



Then substituting in the expressions on page 224, 



Z w = - K'TU, Z p = K'TUy,, Z q = - K'TUx, 



Lt = K'U /moment of inertia of fin relative to the plane y = o] 



Lq = - K'U {product of inertia with respect to x = o, y = o} 



M P = - K'U {product of inertia} 



Mq = K'U /moment of inertia with respect to x = o] 



L w = K'TUy, ; M w = - K'TU Xl . 



A Number oj Small Fins. (General Case) 



T = STi = sum of the separate areas = Total area, x, y, z are 

 the co-ordinates of the centre of pressure of all the fins. 



M lf M 2 , and P are the moments and product of inertia 

 respectively with respect to planes through x, y, z, parallel to 

 y = o, x = o respectively. Then 



L p = K'U{Ty 2 +M 1 }, M p = - K'U{Txy + P} 

 L q = - K'U{Txy+P}, M q = K'U{T5c 2 + M 3 } 



and Z W| Z P , Z q , L w , and Mw hold good as in the last paragraph. 

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