22 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



But the products of inertia (relative to moving axes fixed in the 

 body) D = F = o, because the aeroplane is symmetrical about the xz 

 plane. Substituting the above expressions for h-^, h^, h^, in the 

 equations of motion, and neglecting products of small quantities, 

 we have : 

 du A' , -r I • , n , n\ l- dp E dr J 



^ + Ur= y + g s'mil/ sin (6^ + 6) - g s'lVKji cos (0^ + 6) , 

 dt , 



, Kl'l^^= M+I,T„. 



^-Uq = Z-gcos(0„ + 0). /v-^- ^i^=N. 

 dt i> y "^ dt m dt 



If we substitute for A', Y , etc., their values from the expansion in 

 terms of the first powers of u, v, zv, etc.. and observing that from the 

 conditions of equilibrium. 



Mo + TJi = To + Zo + g sin e^ = Z^-g cos 6^ = o, 

 we will have, making sin 4> — <l>, sin i/' = i/', sin B = B, and cos O—i. 



--^ - uXu + zvX-w + qXq + ge cos 6^0, 



~=qU + uZu + wZ^ + qZq+gd sin 6^, 



^ =-rU + vYv + p Yp + rYr + g^p sin 0^ - g<f> cos 0^, 



Kl^= uMu + wM^ + qMq, 



i^.dl_Edr _ J J r 

 ^'dt Mdi-""^"^^^'^^ " 



We here assume T,, a constant, or that there is no change of pro- 

 peller thrust with small change in forward speed. With a motor in 

 " free route," if the machine speeds up, the propeller tends to race or 

 to speed up so that the slip shall be about constant, and hence the 

 thrust is not materially changed. Since the forward speed (U±u) 

 is approximately equal to U, the thrust is approximately constant 

 and equal to 7"o- 



We have also assumed that T^^ lies parallel to the axis of x. At 

 very slow speed this is not exactly the case and T^ has a small vertical 

 component assisting in sustaining the weight of the aeroplane. At 

 high speeds. To is, however, usually parallel to x and the assumption 



