WILSON— AEROPLANE ENCOUNTERING GUSTS. 245 



where the integrals are loop integrals in the complex plane and 

 I, r], ^ are any functions of A. The results are 



-^ r [(X - ZJ^ - X^rj - {X,\ + g)ne"d\ = P,e-\ ■ 



—. f [- Z„^ + (X - Z..)r? - (Z, + U)nV'd\ = Poe-\ (20) 



—. f[- Mu^ - M,,rj + (y^/X- - M,\)tV'd\ = P,e'^\ 



We next set 



(X - Xu)^ - X^„r, - {X,\ + g)i' = Pi/(X - n), 

 - Z„^ + (X - Zuh - (Z, + U)U = P2l(\ - m), (2i) 



- Mu^ - Mu,v 4- (^ij-X- - il/,X)r = PsKX - m), 

 and solve for ^, r/, ^, finding 



Pi5n + P2hi + i^s^si 



t = 



r = 



A(X - m) 



Pl5l2 + P2822 + ^3632 



A(X - m) 

 Pi5i3 + PoSos + P3533 



(22: 



A(X - m) 

 A = 34( A* + 8.49A3 + 24.5x2 + 3.385A +.917)- 



Bromwich shows that, if with these values of $, ?/, C we take the 

 loop integrals (19) around a very large circle, the results for u, zv, 

 9 will be the solutions for the motion disturbed from rest at the 

 position of equilibrium by the impressed forces P. As he points 

 out, this integration is equivalent to the sum of the integrals around 

 infinitesimal circles about A = /a and about each of the roots A of 

 A^o, that is, the integral is equal to the sum of the residues of 

 ^e^*, r]e^\ ^e^K There is no need to calculate any constants of in- 

 tegration. Moreover any of the quantities u, zv, 6 can be obtained 

 without the others. The numerators in |, -q, C are already calculated 

 in (20 a, h, c) of p. 59. 



