ENGINEERING: E. B. WILSON 295 
Yv, Yp, Yr, Lv, Lp, Lr, Nv, Np, Nr, of which the last nine refer to the 
'lateral' motion which was not discussed. 
The equations for the longitudinal' motion of the free machine are 
du/dt — gd — XuU — XwW — Xqq = XuUi + XwWi + Xqqi, 
dw/dt — Uq — ZuU — ZwW — Zqq = ZuUi + ZwWi -f Zqqi, (l) 
B/M. dq/dt - MuU - MwW - Mqq = MuUi + MwWi + Mqqi, 
where 6 is the inclination of the x-axis, and B/M = is the square of 
the radius of gyration about the 3'-axis. For small oscillations these 
equations are linear with constant coefficients. The solution of (1) 
may be carried out by the ordinary method. In particular the solution 
may be cast in such form that a number of different gusts may be dis- 
cussed very rapidly after a certain amount of preliminary calculation has 
been made. 
The data for the given machine are as follows : 
Xu = -0.128, X^ = 0.162, Xq = Zq = Mu = 0, 
Zu = -0.557, Z«, = -3.95, = 1.74, 
Mq = -150, =34, U = -115.5. 
The numerical equation for u is 
{D^ + 8.49 + 24.5 + 3.385 D + 0.917)w = 
- (0.128 Z)3 + 1.16 Z)2 + 3.385 D + 0.917K 
+ (0.162 Z)2 + 0.715 D + 1.647) Dwi - (59.37 D + 560.6) qi, 
where D = d/dt, with similar results for w and 6, The roots of + 
8.49 + 24.5 Z)2 + 3.385 D + 0.917 = 0 are - 4.18 ^ 2A3i and 
— .0654 =i= AS7i. The solutions are therefore of the form u = e~^-'^^^ 
{A cos 2.43/ + B sin 2.43/) + ^"-'^^^^ (C cos .187 / + sin .187/) + /„, 
where A,B,C,D are constants of integration and lu is a particular inte- 
gral obtained from some particular gust Ui, 'Wi,qi. There are similar equa- 
tions for w and 6. Of the twelve constants of integration only four are 
independent, and the relations between them are independent of the par- 
ticular integrals / (provided the gust is not tuned to the free oscillation). 
It is therefore possible to set down, once for all, formulas for the coefh- 
cients, ^, ^, C, Z> in terms of the initial values of the particular 
integrals lu, Iw, 1$, I'd • Where the machine is in normal flight with 
u=w = B= q=Q. 
The type of gust chosen was / (1 — where / is an intensity- 
factor. This gust rises from 0 to / in an infinite time, but the greater 
part of the rise occurs in the time \/r ox2/r. If we set Wi, Wi, or qi equal 
to / (1 — e~'^), we obtain a head-gust, up-gust, or rotary gust. The 
