WILSON— AEROPLANE ENCOUNTERING GUSTS. 229 



along the x and s axes, a being the angle from the horizontal up to 

 the direction of the shaft. Furthermore if the shaft did not pass 

 through the center of gravity there would be a pitching moment 

 — Phii/U^ if h is the distance of the line of the shaft above the 

 center of gravity. 



28. The equations for the natural longitudinal motion would 

 then be 



{d-Xu + -^A « - X^w - {X,D + g)d = o, (11) 



\ mU" / 



the other two equations remaining unchanged, if we assume for 

 simplicity that a^=h^o. The effect of the varying thrust is to 

 change Xu to Xu — Pg/niU'. We have the value Xu = — .128 for 

 this machine. If the effective propeller horsepower were 87 for 

 U = — 115. 5, the value Pg/mU- is 



Pg ^ 87X550X32 ^ ^^ 

 wZ7- 1800 X 13350 



The modification of the equations of motion on replacing Xu 

 = — .128 by Xu = — .191 would make an appreciable, though not 

 very serious change. 



The determinant A would become 



34Z)* + 290.8D3 4- 850.91)2 -f 165. iD + 31.18 



= 34(D' + 8.553^' + 25.03^^ + 4-856D + .917) 

 as compared with 



S4(D' + 8.490/)^ + 24.501)= + 3.385D + .917) . 

 The rapidly damped oscillation would, as a first approximation, be 



— 4.276 ± 2.5961 instead of — 4.245 ± 2.545/. 

 The first approximation for the small root would be 

 — .097 ± .1771 instead of — .069 ± .i8ii. 



The damping would be more pronounced and the oscillation a trifle 

 faster. 



29. It may be concluded that whether the screw is supposed to 



