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SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 62 



The right-hand members of these equations are no longer zero if 

 any wind gusts are assumed.' The complementary function may be 

 found by the well-known " operational method '" by algebraic solution 

 for D. (See: Wilson's " Advanced Calculus,'' p. 223.) 



The physical condition that the three equations shall be simul- 

 taneous is expressed mathematically by equating to zero the determi- 

 nant A formed by the coefficients of the variables u, zv, and 6. Thus : 

 Z) + X«, ~X^, -(XqD + q) 

 ~Z,„ D-Z^, ~(Zr,+ U)D =0. 

 -Mu, -M^o, (KID'-AUD) I 

 Expanding the determinant we obtain : 



where for abbreviation : 

 A,=K%, 

 B^= - (M, + X„ICs + ZuJ<%), 



Am, a to, Ao 



C,= 



+K: 



Am, Xi 



D,= - 



E,= 



Mu, My,, 



Am, At< 

 7 7 



Mu, M 



Q 



U + Z^ 

 M 



Q 

 Q 



cos Of, 

 sin df, 

 o 



Mu, 



( - ) sin 

 cos 0O 



The solution of the biquadratic A for D is of the form : 

 D — a, h, c, or d, 



where A'j, K^, K^. K^, Kr„. . . ./v-l2 ^re constants determined by initial 

 conditions. Solutions for u and zv are similar. 



The condition for stability of motion is that 6, u, and zv shall 

 diminish as time goes on. Hence, each of the roots of the biquadratic 

 must be negative if real, or, if imaginary, must have its real part 

 negative. This condition for stability may be applied without finding 

 the constants /v^ to K^^, by solving only the biquadratic for a, b, c, d. 

 Indeed, Bryan has shown that by use of Routh's discriminant the 

 biquadratic need not be solved. The condition that a biquadratic 

 e({uation have negative real roots or imaginary roots with real parts 

 negative, is that A-^, B.^, C^, D^, E-^ and B ^C^D ^ — AJD ^^ — B ^^E^ be 

 each positive. 



^ Loc. cit.. p. I, §1, footnote 3. 



