NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 21 



In the theory of small oscillations ti, v, w, p, q. r are small by hy- 

 pothesis and we may expand X by Alaclanrin's theorem, neglecting 

 squares and products of these small quantities. Hence, 

 X = Zo + uX,, + vX, + ■zvX.o + pXp + qXq + rXr, 

 Y=Y^ + uYu + vyv + -ci'y,o + pYj, + q\\ + rYr, 

 and similar equations for Z, L, M, N. 



Here A'„, A'r, etc., are the partial derivatives of .Y with respect to 

 u, V, etc., and are the rates of change of A' with ii. v, etc. That is, 



" dV 'u ■ 

 There are, therefore, 36 " resistance derivatives '" involved which 

 are constants for the aeroplane and depend upon the arrangement 

 of surfaces and their presentation to the relative wind. 



Fortunately, for reasons of symmetry, 18 of these derivatives 

 vanish, for example : Xv, Xp, X,-. We then write : 



Z = A^o + w Xm + wXw + qXq, 



M = AIo + t{Mu + zvMy, + qZq, 



Z = Z(, + uZu + wZui + qMq, 



Y=Yo + vYv + pYp + rYr, 



L = Lq + vLv + pLj> + rLr, 



N = No + vNv + pNp + rNr. 

 The above expressions are only approximate if ii, v, w, etc., are not 

 small. 



The equations of motion for a rigid body having all degrees of 

 freedom, are : 



j" +wq-vr = X + T, + gsm{6,-¥e), 

 '!lf +(^ + ^')r- zvp =Y-g sin <^, 

 ^-^vp-{U + u)q = Z-gcos{6, + 6), 



at - ' -f 3 



^-- -ph:i + rh^ = niM + h T^, 



/here 



-^ -qh^ + pho=:mN, 



h^ = pK \m — qF — rE, 

 h., = qK'jiiii — rD — pF, 

 /13 =: rKl'in — pE — qD. 



