NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 25 



In a similar manner the equations (ib) defining the asymmetric 

 motion may be expressed as Hnear differential equations with con- 

 stant coefficients. 



Substitute D-d> for ^/ and Dd) for h.' Then : 



(D - Yv) V + ( U - Yr)r +{g- VpD ) </> = o, 



- L,v - L rv + ( K\D- - LyD )cj> = o, 



- Nr:V + ( KID - Nr ) ,' - .V,,Z)<^ = O, 



A., = A,D' + B.D^ + C.D- + D,D + E., = o. 

 where : 

 A,=K%K\, 



C.^-LrX. + NrL. + KlLpYr + KrYvKl+XrUKl 



-{L,YpK% + N,YrK\), 

 D,= Yr{LrNj>-NrLp) + L,{UNp+ gK%) - ULj.Nv 



+ (N,-YrL,-L,Y,N„ + LrY,Nr~N,YpLr), 

 E,=g(NrLr-L,Xr). 



As before, the condition for stability is that the real roots and real 

 parts of imaginary roots of the l)i([ua(lratic be negative. 



§10. CONVERSION TO MOVING AXES, LONGITUDINAL DATA 



Horizontal flight at 0° incidence i of wing chord re([uires a 

 speed of 112. 5 feet per second, or about // miles per hour (see the 

 characteristic performance curves). The normal attitude then has 

 the axis of x parallel to the wing chord and horizontal. The axis s 

 is vertical. For slow speed with an angle of incidence i of 12°, a 

 speed of 54 feet per second, or about t,/ miles per hour, must be main- 

 tained. In this case, the normal attitude has the axes x horizontal 

 and c vertical, but the axes are entirely different from those used for 

 tlfe high-speed condition if they are considered with reference to the 

 aeroplane. The axis of v is, however, the same in both cases. 



' Since we consider only the small oscillations, </> and i^ are of the nature of 

 infinitesimals, and hence compound vectorially as do p and r. Professor 

 E. B. Wilson suggests the important implification of the treatment given by 



Brvan or Bairstow due to making j — /> and -j^—r. They used angular 



at at 



coordinates giving expressions for j^and-jy in terms of p and /- and the 



angles which are initially cumbersome but ultimately reduce to the simple 

 form here given. 



