ATMOSPHERIC FRICTION. 
267 
transportation, and particularly the cost of propulsion in 
aeronautics. Let us consider the soaring plane, first assum¬ 
ing it smooth, then frictional. 
Let A be the area of the plane, W its weight, v its veloc¬ 
ity, a its angle of flight, R its resistance, H the propulsive 
power, and ? the density of the fluid in which it is moving. 
Then, if the plane is 'frictionless and steadily soaring on 
a horizontal course in still air, 
R — W tan a, ..(a) 
H=^ Rv, .. (6) 
W = 2 Ic9 Av 2 sin a COS a, ... (c) 
1 + sin 2 a 
the last expression being the lift as given by Duchemin’s 
formula, in which k is a constant of figure. 
The relations of these seven variables contain much that 
is of interest in the theory of the aeroplane. For example, 
let us find the mileage cost and the propulsive power when 
the plane is just soaring. 
The mileage cost is proportional to the resistance divided 
by the load, and hence, as shown by equation (a), it is 
directly proportional to the tangent of the angle of flight. 
It may therefore have any value, from zero to infinity, 
according to the inclination of the plane, and if this be kept 
constant the mileage cost is the same for all velocities, for 
whatever extent of surface, and for all densities of the 
medium, from mountain air to sea water. 
In a similar way the mileage cost may be studied as a 
function of any of the other variables. Thus from equation 
(c) we obtain 
tan ' c _ W{l+sin>a) 
in which the ratio of the parenthetical factors is practically 
unity for small values of «. Hence, writing 
tan a = 
W 
2 k? Av 2 ’ 
