68 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



tion from normal attitude and to correct it by use of his controls. 

 However, the aeroplane could only be flown at this speed even in still 

 air provided the pilot were alert. 



The second factor is a strongly damped subsidence D= —g.i2, 

 which damps to half amplitude in .08 second. 

 The third factor is an oscillation, 



D- + .2^iD + .292 = o, 

 £>= '-.ii6±.528f, 



having- a period of "^5= 12 seconds, which is damped to half ampli- 



.52b 



tude in t= ^-^ =6 seconds. This oscillation is stable, but the damp- 

 .116 '^ 



ing is only moderate, and it may well be felt on the aeroplane in flight. 



In some types of aeroplane, it is likely that this motion may be 



undamped and hence the amplitude of successive oscillations will be 



increasing, giving rise to instability of a new character. 



§11. THE "SPIRAL DIVE" 



The motion found corresponding to E.y negative, as at slow speed, 

 may be traced to the resistance derivatives involved in the expression 

 for E.,. Thus : 



E. = g(N^Lr-L,Nr), 



and Eo will be positive only when Lv/Nv is greater than Lr/Nr. For 

 stability, or E.^ positive, L^ and A^,- should be large and Nv and L,- 

 small. 



The derivative Lv depends on the rolling moment due to side slip 

 and can be made large and positive by an upward dihedral angle to 

 the wings or by vertical fin surface above the center of gravity of the 

 aeroplane. At low speed and high angle of incidence we see that Lv 

 is diminished. Thus, at 6° and 44.6 miles, L„ = 3.44, while at 12° and 

 36.9 miles, L, = i.9i. The drop in speed is only about 18 per cent. 

 Hence the drop in L,j cannot be due to the lower speed, but must be 

 due to the greater angle of incidence. 



Let i be the angle between the wind direction and the center line of 

 the wings where yaw ij/ is zero. Let cj> be the angle through which 

 each wing tip is raised, and let the angle between the wind direction 

 for a yaw 1/' and the plane of the chord of the up wind wing be i'. 

 Then it can easily be shown by geometry that approximately 



when i, \p, and /? are small ^ and expressed in circular measure. 



^A. Page, "The Aeroplane," p. 82, Griffin, London, 1915. 



