66 Proceedings of Boycd Society of Edinburgh. [sess. 
The nett force acting upon the bird, owing to its motion through 
the air, should be treated as consisting of two elements, viz. (1) 
the normal force on the wings due to their obliquity a to the line 
of motion, (2) the resistance due to the air friction on the wing 
surfaces and body. Taking, then, 
A = total wing surface (one side) ; sq. ft. 
rA = total surface area as reckoned for computing resistance; do. 
Y = speed through the air ; ft. per. sec. 
These two elements of force may be expressed thus — 
Normal force (lbs.) = PAW , 
Resistance (do.) = FrAY 2 , 
[where P and F are constants appropriate to the resisting medium, 
in this case air]. 
The strict condition of equilibrium for constant speed and direc- 
tion is that the resultant of these forces should be vertical, and 
equal to the weight of the bird. Seeing that the values for a and 
a with which we have to deal are small, this condition is defined 
with sufficient approximation for our purpose by the equations — 
FrAY 2 
Wa — FrAY 2 ; whence a — ^ ; 
(i) 
W 
W = PAYV; whence o- = p^y 2 ; 
(2) 
[where W = weight of bird in lbs.]. 
In order that the soaring may take place without the bird losing 
level, the air must have an upward motion (or upward component 
of motion), the speed of which (with a similar approximation) may 
be expressed as — 
= V(a + cr) , 
FrAY 3 W 
W + PAV’ • • • • (3) 
By equation (1), the speed Y depends on the wing angle a, which 
the bird may regulate at his pleasure ; and we will assume that he 
thus assigns such value to Y ( = say Y x ) as gives minimum value to 
V(a + o-). By differentiating equation (3), this value is determined 
as — 
