HES “vee 
NAVAL ARCHITECTURE TO AERONAUTICS. 289 
Wi 
where 7= 10° we have Fa cos 4=KAV'l sini or KA vat cot 1=1.35W. 
Substituting in (3) Px=1.35 WI sin 7. 
If the inclination is changed the righting moment will be the difference 
between the moments of the weight and the moment of the pressure whence: 
1.35 Wisin i—o0.25W1 cost= WI (1.35 sini—0o.25 cos 7). (4) 
If @ be the angle of incidence at the position of equilibrium and this be 
increased by a small angle, a, we shall have from (4) the righting moment. 
M=W/1 [1.35 sin (+a) —0.25 cos (0-+a).] (5) 
If GM be the metacentric height of the plane at the angle 0, then the 
righting moment when further inclined by the small angle a, will be: 
GMXW sin a= W/1 [1.35 sin (@+a)—0.25 cos (@+a).] (6) 
(or) GM =I (1.35 cos 6+0.25 sin 6). (7) 
Under the conditions assumed 0 is approximately 10 degress. Using the 
value in (7) gives a metacentric height 
GM = 1.371. 
For a plane 6 feet in depth and of the usual width of an aeroplane this 
would give a longitudinal metacentric height of about 8.2 feet. 
It is to benoted, however, that the righting moment is not equal for equal 
changes up and down from the position of equilibrium. This will be seen 
from an inspection of equation (3). Increase in the angle of inclination 
causes an increase in the total pressure and results in a larger righting moment 
than is the case for decrease in the angle of incidence. 
This conclusion seems to be confirmed by experience with actual aero- 
planes which have failed because of insufficient fore and aft stability. They 
fall head first and not tail first. 
With regard to transverse stability of the plane considered above it is 
obvious that transverse inclination will produce no change in the transverse 
position of the center of pressure and as the center of gravity is assumed to 
be at the center of pressure, there is no righting moment to resist such incli- 
nation. The transverse stability is nil and if inclined to one side or another 
the plane will fall sidewise. This would be modified somewhat, however, by 
the resulting transverse velocity having some tendency to right the plane. 
