NO. 5 STABILITY OF AEROPLANES IIUXSAKER ANP OTHERS 23 



that it al\va\s is so parallel is here made for sinii)licity. Tn any case 

 To is eliminated l)y the conditions of ecjuililjrium. 



In the present investigation the normal flight path is assumed 

 horizontal, or ^„ = c>- The product of inertia E is small for ordinary 

 aeroplanes with the heavy weights fairly symmetrical above and 

 below the axis of x. In view of the probable insignificance of E and 

 the fact that E cannot easily be determined for an aeroplane by 

 simple experiments, it is here neglected. In the simplified form the 

 equations of motion then are : 



^ = uA\ + wX,„ + gA', + ..'^. (la) 



^ = qU + iiZ, + zvZ,o + qZq, (la) 



dv 

 ~dt 



^-rU + vYv + pVp + rYr, (ib) 



K,^=i'E, + pLp + rEr. (lb) 



K%-^=uMu + zvM,, + qM„ (la) 



Kl^ =vN, + pNp + rNr. ( ib) 



It is seen that equations (la) involve only the longitudinal motion 



or motion in the plane of symmetry' .r^c: of the aeroplane, since p, r, v, 



and ^ do not appear. Likewise, equations (ib) involve only the 



asymmetrical motion, lateral and directional, and do not contain 



6, u, u% and q. The two sets may then be considered separately, the 



former on integration giving' the "symmetrical motion" and the 



latter the " asymmetrical motion." 



d6 

 Since -j- =q, eciuations (la) mav be written in terms of three 

 at 



variables u, -a', and H and their first derivatives. The " resistance 



derivatives '' A'„, A',., A'^, etc., are constant coef^cients. The three 



variables are each functions of the time, and the three equations at 



any instant of time must be satisfied by a concordant set of values of 



u, zv, and 6. The equations are, therefore, simultaneous and .are 



linear differential equations with constant coefficients. 



Writing the operator D to indicate differentiation with regard to 



d 

 tune or , 



{D-x,)ii-x,,zi'-(x,D+g)e=o, ^ 



-ZuU+{D-Z^)iv-(Zq+U)De = o,[ (2a) 



- Muu - M^w + ( Kb-D' - MgD )e = o. j 



