On Flapping Flight of Aeroplanes. 421 



horizontally, and be supported by wings, consisting of planes of 

 negligible mass, held nearly edgewise to the direction of motion, and 

 moved vertically up and down by some machinery carried by the 

 heavy mass. Let there be in addition some arrangement of the 

 machinery by which the angle of inclination of the planes or wings 

 to their direction of motion is variable. We shall suppose the velocity 

 of progression high, and variations of the propelling force small 

 enough for changes of forward velocity to be neglected, compared to 

 the average forward velocity. This is justified by Langley's experi- 

 ments, which show that, for small inclinations of an aeroplane, the 

 direct resistance to forward motion is small compared to the supporting 

 force. We shall also, in the first instance, neglect direct air resist- 

 ance on the heavy mass as small compared with that on the aeroplanes ; 

 and, for convenience, we shall call the heavy mass the bird's body 

 and the attached aeroplane the wings. 



The case differs from that of a real bird most notably by the circum- 

 stances that the aeroplane in our theoretical case has no mass, whereas, 

 a bird's wings have one of very Sensible magnitude compared with 

 that of its body, and the aeroplane is supposed to be moved up and 

 down, relatively to the mass, as a whole, instead of being pivoted to 

 it, as a bird's wings are to its body, besides being of constant area, 

 which wings are not. Consequently, as before remarked, numerical 

 results can only be regarded as indicating the order of magnitude of 

 the quantities calculated, not their exact values. 



The inclination of the wings being variable, and their motion being 

 compounded of an up-and-down one with a forward one, it is evident 

 that the supporting force may be periodically variable also, and con- 

 sequently the bird's body will move in a sinuous or wavy path, on the 

 whole horizontal, and the stroke of the wings will be the relative 

 motion of wings and body. If the horizontal velocity be high, and 

 the amplitude of the wing stroke relatively to the air moderate, and 

 the variations in the supporting force not too great, the path of the 

 bird's body will be only slightly waved, so that fig. 1 may be taken as, 

 in a general sort of way, representing what takes place. 



In the same figure AiBi, A 2 B 2 , &c., represent the plane of the wing 

 at different successive positions, seen edgewise, and P l5 P 2 , &c., the 

 corresponding resultant air pressures on it, in direction and magnitude. 

 Now it is plainly a matter of arrangement of mechanism what these 

 shall be, as they depend on the angle between the lines AiBi, &c., 

 and the tangent to the heavy line, marked " Path of wing " at each 

 instant. Consequently it is worth while inquiring what are the con- 

 ditions of adjustment to cause a forward force on an average to be 

 applied to the bird and wings, when the wings are moved up and down 

 by an engine forming part of the bird's body, when the supporting 

 force is, on an average, equal to the bird's weight, and the forward 



2 K 2 



