244 SYMPOSIUM ON AERONAUTICS. 



Much may be said for their method of expansion in series — for 

 some problems the work is decidedly simpler than with my method. 

 It has been my experience, however, that the application of series 

 to the motion of any aeroplane has its own difficulties and com- 

 plicated calculations when the motion is to be followed for any 

 reasonable length of time and especially if the machine is defined, 

 as I have always preferred to regard it as defined, by the actual 

 coefficients determined by wind tunnel experiments rather than as 

 Bryan's skeleton plane consisting of a main front plane plus tail 

 plane, — even though the results obtained from such a skeleton may 

 be extended to more complicated machines by Bryan's invariant 

 method (see his "Stability in Aviation," Chap. VI.). 



49. The question therefore arises whether there may not be 

 some way of abridging the calculations leading to the actual motion 

 of the machine. Since finishing my work above, I have received the 

 Proceedings of the London Mathematical Society, 15, 1917, Pt. 6, 

 in which there is an article on " Normal Coordinates in Dynamical 

 Systems," by T. J. I'A. Bromwich in which he develops a method 

 of treating the motions of dynamical systems by means of the theory 

 of functions of a complex variable. I wish, in closing, to describe 

 the application of Bromwich's work to the problem in hand. 



We have to solve for the longitudinal motion equations of the 

 type 



(D — Xu)u — X^,^v—(XqD+g)0 = P,e'^\ 



Z„u-}~(D — Zu-)w—(Zq^U)De = P,e'^' (18) 



— Muu — Murcv + ( kJ'D" — MqD ) 6 = P,e<^ *, 



where /x is a real or complex number, the values we have used being 

 o, — r, ± pi. We substitute 



u = —.\ e"^d\, 



27rt Jo 



= -^ feVX, (19) 



= — fe"i'd\ 



27njJo 



w 



