1890 - 91 .] Mr R. E. Fronde on the Soaring of Birds. 
67 
4/3PF? 
• [- 
76 x 
0) 
whence, by substituting in (1), (2), and (3), we get for the corre- 
sponding values of a, <r, &e., = say a x , o-j, &c., as follows : — 
-yaw?; • • • • 
a 1 + <r 1 ( = 4a 1 ) = 2-3iy^ ; . . . 
(5) 
(6) 
(7) 
( 8 ) 
Even without putting numerical values to the symbols, we may at 
once note some interesting conclusions which follow from the struc- 
ture of these equations. 
(1) The angle values a l5 oq, and (cq + oq), bear a constant ratio to 
one another, independent of the values assigned to F, P, r, A, or W. 
Fr • 
(2) These absolute angle values depend solely on _ , (in which 
F and P are constant for a given medium, and r dependent only on 
the proportions of the bird), and are independent of ^ , or weight 
A 
per square foot of wing area. 
W 
(3) Hence, — influences the value of V 1 (a 1 + oq) only as influ- 
A 
encing the speed. 
(4) In similarly proportioned birds of different since since 
W 
(or weight per square foot wing area) varies as dimension, Y 1 (a 1 + oq) 
varies as square root of dimension. 
This expression, V^cq + oq), besides indicating the minimum 
speed of ascending air current necessary for soaring or quiescent 
flight, without loss of level, furnishes also a presumable measure of 
the minimum effort needed to sustain active flight in still air with- 
out loss of level. For in maintaining his level by active flight, the 
bird must be supplying at least the work theoretically equivalent to 
lifting his own weight at the rate at which he would quiescently 
