WILSON— AEROPLANE ENCOUNTERING GUSTS. 225 



To put it another way, there is a discontinuity in the stress at the 

 initial instant — as might be expected. The amounts of the discon- 

 tinuities are also just what might be expected, viz., X„/ and Z,,/. 

 In like manner for an up-gust the initial discontinuities in accelera- 

 tion are X^J and Z^J along the x and z axes. These results could 

 have been foreseen from the differential equations themselves as 

 well as from " common sense." The path in space is not materially 

 different for an infinitely sharp gust from what it is for a reason- 

 ably sharp gust. 



It may therefore be said that a tolerably good idea of what 

 happens for sharp gusts may be had from the consideration of 

 infinitely sharp gusts. 



24. It has just been stated that the conclusions concerning the 

 initial accelerations may be foreseen from the differential equations. 

 This may be proved as follows : We have 



{D — Xu)u — X^^zi' — Xqq — 5^(9 = Xuih + -Yu-w^ + X^q^, 



— ZuU-i- (D — Z,o)-w — (Zq-\- U)q=^ZuU-, -^ Z ^w^ -{- Z qq „ 



— MuU — MuW + (k^W — Mq) q = Mutt, -f Mr^zu^ + Mqq„ 

 De — q = o, 



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where the equations have been reduced to four involving only the 

 first derivatives of the four variables u, w, q, 6, with the initial condi- 

 tions w = 'zt'=g = ^ = o, by the device of choosing q = D9 as an 

 independent variable so as to eliminate the second derivatives. 



These equations determine the first derivatives at the initial 

 instant or at any instant in terms of the values of the variables at 

 that instant, namely. 



Du = XuU + X,cZO + Xqq ^ gO -\- XuH^ + XicZV^ + Xqq^, 

 Dw = Zuti -f Z.cZV + Zqq -\-Uq-\- ZuU^ -\- Zy^w^ -{- Zqq^, 



k/Dq = MuU + Mtcxv -f AL.q + MuU^ + Mwzv^ + Mqq^, 

 D6=q. 

 At the initial instant u, zv, q, 6 vanish. 



PROC. AMER. PHIL. SOC, VOL. LVI, P, JUNE 21, I917. 



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