210 SCIENCE PROGRESS 



and canted through <£ (see Stability in Aviation, pp. 20-27) we 

 have by the theory of moving axes : 



— f-Tj- + qw - rv ) = accelerating force along Gx = Wsintf + H - X 



W/dv \ 



"gAdT + ru "" pw J = " " " Gy = W cosecos( i> ~ Y 



Yldf + pv ~ qu ) = " " " Gz = " Wcos sin( £ - Z 



A. ^P _ Z_ ^9. . c - B ESL . F EI = ( accelerating moment of \ Gx _ _ L 

 g dt " g dt ' g g "I forces about / 



g dt gdt T g g 



gdt T g g 



where X, Y, Z, L, M, N are the components of air resistance, 

 H the propeller thrust acting in general parallel to the axis Gx 

 at a perpendicular distance h below. For simplicity F may be 

 taken to be zero, in which case the axes are the principal axes. 

 To X, etc., may be added terms X u etc., due to gusts of wind. 

 We start with the assumption that X 1( etc., are zero — i.e. that the 

 air is still. 



Assuming that the plane is descending uniformly at an angle 

 6 with the horizon so that </> = o, i.e. it is upright, u = v = p = 

 q = r = w = o initially. On disturbance they represent small 

 increments. U is the steady velocity forward. 



Let X , Y , etc., represent the initial resistances. 



.-. O = Wsin0 o + H - X = Wcos0 o - Y = Z Q 

 = - L = - M Q = - H h - N 



these being the equations of equilibrium in steady motion. 

 Where H is inclined at an angle i) with the direction of flight, 

 the first two equations reduce to 



O = Wsin0 o + H cos>7 - X 

 O = Wcos0 o + H sin»7 - Y . 



Now let the aeroplane be disturbed and the above increments 

 enter. The resistances are functions of the velocities, and we 

 have — neglecting squares of small quantities — 



X = X + X u u + X v v + X r r + X w w + X p p + X q q 



with similar expressions for Y, Z, L, M, and N. 



