80 
Proceedings of Boyal Society of Edinburgh. [sess. 
rate of the wind continually varies (as is usually the case), it may 
substitute for the string the resistance due to its own inertia. 
We need only assume for the bird, a power of balancing itself at 
any inclination, so as to be able to present the area of its wings to 
the wind in the most advantageous position. 
Suppose the wind blowing from right to left in fig. 1. 
Let EF represent the trace of the surface which the bird creates j 
area A square feet. Then, if EF is inclined to the horizontal at the 
angle a, the area obstructive to the wind is A sin a, and the mass 
of air obstructed per second 
GA sin a 
= u 
9 
u being the relative velocity of the wind. 
This produces a pressure at right angles to EF proportional to 
the resolved velocity in that direction 
GA si 
sin a 
u{u sin a) or 
GA sin^ a 
~9 
Resolving this pressure vertically and horizontally, we obtain a 
Cx-A. sm^ CL 
vertical force id- cos a lbs. to support the weight of bird, 
GA 
and a horizontal force — ?^sin^a producing acceleration in the 
direction of the wind. 
Let us suppose the bird to be just supported. Then equating the 
former force to W, the weight of the bird, we have 
sin^ a cos a = W 
g _ 
from which 
GA sin^ a cos a 
As the bird is supposed to be able to poise itself in the most 
advantageous way, we must determine what value of a correlates 
with the smallest value of u. 
Differentiating the expression for and equating the derived 
coefficient to zero, we get 
2 sin a cos^a = sin^ a 
or _ 
tana= ^2 
so that the most advantageous inclination is tan J'2 or 54° 44'. 
