WILSON— AEROPLANE ENCOUNTERING GUSTS. 241 



for / are likely to arise in flight under normal conditions. In so far 

 as experience shows that machines are not liable to roll and side- 

 slip, it is pretty good evidence that aerial rotary motion with axis 

 parallel to the earth is rare and small. 



41. Case j. — Yawing gust. r^ = /(i — e'^). 



/,,// = 25.67^-*, 7,0 = 25.67/, 



/,// = — I — . 1235^-^ 7,0 = — .1.12/, 



Co^ = — .006/, Co2= — .oo67, ^23 = — .06347, C.^ = .0702J. 

 In this case the motion is 



^/J = — .oo6r'^--'2*'^ — .006^-^-^*^^ — .0792 -f . i54£?-* 



_j_ ^-.326Si(^ — .0634 cos 1.215^-}- .0701 sin 1.215O, 

 v/J = + .05^0"*^*'' — .02y-^-^*-^ + 25.67^-' 



_j_ ^-.326st(- — 26.35 COS i.2i5f-[- 108.9 sin 1.215O, 



r/J=^ — I — .1235^-* -f c--^-*'^*(i.i45 cos i.2i5f-f .222 sin 1.215?). 



For moderate values of J, there is nothing serious indicated. The 

 coefficients of the divergent terms are small. There cannot be much 

 roll. The most noteworthy phenomenon is the large amount of 

 side-slip which is fairly rapidly damped out. 



42. This leaves the rolling gust as the only dangerous type of 

 lateral gust. 



The infinitely sharp side-gust would produce an initial accelera- 

 tion YJ. 



Constrained Aeroplanes. 



43. Suppose now that by some automatic steering device the aero- 

 plane were constrained to remain pointing in the same direction, 

 i. e., so that r = o identically. The equations of motion become 



(D-Y,)v-^(g-YpD)cj> = Y,v,-^Y,p,-^Yrr,, 



— L,v-\-{k/D — L,)Dcf> = L,v^ + L,p,-{-Lri\, (15) 



— N,v—NpDcf> = N,v, + Npp, -f Nrr, + F, 



where Fin is the moment necessary to maintain the constraint. The 

 last equation may be regarded as determining F. 



PROC. AMER. PHIL. SOC. , VOL. LVI, Q, JUNE 20, I9I7. 



