138 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 1908. 



Comparing this value of ku with Langley's value 0.004 for a plane 

 surface V being in miles per hour, we see that the lift for the arched 

 surface is 25 per cent greater than for a plane surface of the same 

 projected plan. That is to say, this arched surface is dynamically 

 equivalent to a plane surface of 25 per cent greater area than the 

 projected plan. Such a plane surface may be defined as the " equiva- 

 lent plane." 



Resistance and propulsion. 



The resistance of the air to the motion of an aeroplane is composed 

 of two parts, {a) the resistance due to the framing and load; (5) the 

 necessary resistance of the sustaining surfaces; that is, the drift or 

 horizontal component of pressure, and the unavoidable skin friction. 

 Disregarding the frame and considering the aeroplane as a simple 

 plane surface, we may express the resistance by the equation 



R = W tan a + 2/A (3) 



in which E is the total resistance, W the gross weight sustained, a the 

 angle of flight, / the friction per square unit of area of the plane, A 

 the area of the plane. The first term of the second member gives the 

 drift, the second term the skin friction. The power required to 

 propel the aeroplane is 



H = EV 



in which H is the power, V the velocity. 



Now W varies as the second power of the velocity, as shown by 

 equation (1), and / varies as the power 1.85, as will be shown later. 

 Hence we conclude that the total resistance E of the air to the 

 aeroplane varies approximately as the square of its speed, and the 

 propulsive power practically as the cube of speed. 



Most advantageous speed and angle of fight. — Again, regarding 

 W and A as constant, we may, by equation (1), compute' a for various 

 values of V, and find / for those velocities from the skin-friction table 

 to be given presentl}^ Thus a, E, and H may be found for various 

 velocities of flight, and their magnitudes compared. In this way the 

 values in Table 1 were computed for a soaring plane 1 foot square, 

 ■ weighing 1 pound, assuming ha = 0.004, which is approximately 

 Langley's value when V is in miles per hour. 



