ATMOSPHERIC FRICTION. 
269 
tion for the speed of fall of a horizontal plane haying lateral 
motion. If v x be its edgewise speed, v y the speed of fall, 
then its true speed, v, equals \/ v * -f^ 2 , and the angle «, be¬ 
tween v and the plane, is determined by the equation 
tan a — v y / v x . Substituting this value of a in equation ( d ), 
we have 
v y __ W (1 + sin 2 a) 
v x 2 k$ Av 2 (1— §in 2 ay 
which, for high speeds and moderate loading, becomes, 
_ W (1 + sin 2 «) 
Vy 2 ks Av 
since « is small, and v x is nearly equal to v. Under these 
conditions the speed of fall varies inversely as the speed of 
flight, which means that the rate of descent and the power 
expended may be made indefinitely small by sufficiently in¬ 
creasing the speed. Of course, if the air has an upward 
trend equal to or greater than v y the plane will soar con¬ 
tinuously on a horizontal or ascending course. 
Suppose the gliding plane to dip a degrees below the 
horizon, and to have a forward resistance. The angle of 
impact of the air is d = # — a, in which tan & = v y / v x , as 
before; and, when steady motion is established, the hori- 
, i 2 ks Av 2 sin d sin «, just 
zontal component of the air pressure,-^^^ 
r 1 + sm 2 d 
equals the horizontal resistance. Accordingly the plane 
will glide continuously with the constant component veloci¬ 
ties, v x forward and v y downward. If, however, the air has 
an upward trend equal to v y , or greater, the plane will glide 
continuously on a horizontal, or ascending course. This is 
the principle of one kind of soaring practiced by the birds.* 
*The Wright brothers report that they can glide continuously down 
a seven-degree slope at a speed of 18 miles an hour in still air. This 
means that if the air has an upward trend of 18 X sin 7° = 2£ miles an 
hour, they can glide on a horizontal course indefinitely at a speed of 18 
cos 7° = 17.06 miles an hour. Hence in a soaring pavilion having a forced 
