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SCIENCE PROGRESS 



X is therefore given by 



Wj+X a 



X. 



w 



x r - x cos ^> 



wx „ „ wu w . . 



~^ + Y v Y r + — + T sintf fl 



N. 



CX 



+ N r 



= O. 



Multiplying the last column by \ we have on expanding an 

 equation of the fourth degree in X. 



Asymmetrical Oscillations l 



<£cos0 o = pcos# — qsin0 o 

 but 



<£= -^ = \<p (assumed) 

 .'. \<f>cosd = pcos# - qsin0 o . 



The equations 3, 4, and 5 on p. become 



wu w . a \ n 



in0 o Jq = O 



Gj + Z w)w + (^ cos* + Z p ) p + (Z c 



su 



g 



L w w + (A^+L p )p + (-F^ + L q )q = 

 M w w+(M p -F^)p + (B^ + M q )q = 0. 



Again we have an equation of the fourth degree to find \ by 

 expanding the following : 



WX _ _ , w a „ WU W . a 

 — + Z W , Z p + T cos^ , Z q --j^- T sm6 a 



M 



w 



A X T 



g P 



- F- + M 



g 



P> 



g q 



B- + M q 

 g q 



= O. 



In both types of oscillations we have u, v, w, etc., of the form 

 a^' 1 + a 2 e A -* 1 + a s e A » l + a 4 e^ or 2(a s e* st ). 



For stability then X s must be such that the real part is negative 

 or zero. If the real part is positive there is instability. 



For stability \ = - a s ± i& {a s = or > o}. If & is zero 

 there is subsidence, and if & is not zero there is oscilla- 

 tion and subsidence, two of the terms 2(a s e Ast ) reducing to 



e~° st (acos/3 s t + bsin£ s t) where a and b are arbitrary constants. 



1 See Stability in Aviation, p. 31. 



