THE DISTURBED MOTION OF AN AEROPLANE 223 



Symmetrical Derivatives 



C is an arbitrary point (xy). P is the centre of pressure, so 

 that CP = a<f)(a) (a being small) and R is the normal thrust. 



>x 



direction of 



flight —>* 



The aeroplane receives increments u, v, w, p, q, r, so that 



v +xr 



8a 



U 



R = KS(U + d\i)H(a + 8a) = KSU 2 f(a) + ^ Su + ~ 6a 



= KSU 2 f(a) + 2KSUf(a)(u - yr) + KSUf(a)(v + xr) 



v -4- xr 

 £ = xcosa - ysina + a<£ (a) + a$'(a) . — =-= — . 



Prof. Bryan also finds that due to the rotation "r" of the 

 plane about C, f(a), <£(a) are functions of — . He calls 



f rW (=^.u)and*r W (=^)u 

 the rotary derivatives. 



X = Rsina, Y = Rcosa, N = R£. 



We then find that 



X Q = KSU 2 f(a)sina, 

 X u = 2KSUf(a)sina, 

 X,; = KSUf'(a)sina, 



Y = KSU»f(a)cosa, 

 Y u = 2KSUf(a)cosa, 

 Y v = KSUf'(a)C0Sa, 



X r = KSU{xf'(a) + f r (a) - 2yf(a)}sina, Y r = KSU{f r (a) + xf'(a) - 2yf(a)}cosa. 

 N = KSU>f(«)£, N v = KSU {f'(a)£ + af(a)(/)'(a)}, 

 N u = 2KSUf(a)£, N r = KSU{f'(a)x - 2yf(a) + f r (a)j£, + 



KSUf(a) . {xa(£'(a) + a«/) r (a)j. 



See Prof. Bryan's work (p. 41). 



