4 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL, 144 



of the wing length, we must conclude that wing thickness increases 

 with the 1.34 power of the wing length and that the wings include a 

 steadily increasing percentage of total weight as the size of the animal 

 increases. 



While we know little about the structural properties of bird and in- 

 sect wings, it is reasonable to assume that if the thickness increased as 

 the first power of the length, the angular deflection at the wing tip dur- 

 ing, let us say, the downbeat would be constant. Since wing thickness 

 actually increases as the 1.34 power of wing length, the angular deflec- 

 tion at the tip must decrease with increasing size (or weight) of the 

 animal. This may be related to maintenance of aerodynamic efficiency 

 with increasing size, but the argument is certainly not an obvious one. 



It is even more extraordinary to note that the data for insects and 

 birds fall on a continuous straight line. The materials of which the 

 wings are constructed are totally different for the two classes ; a ribbed 

 chitinous membrane for the former and a complex structure of bone, 

 muscle, and feather for the latter. It must, however, follow that the 

 mean density of wings remains the same quite regardless of the mate- 

 rial of construction. 



It follows from the wing area-wing weight relationship that the 

 weight of the wings will comprise a steadily increasing percentage of 

 total body weight as the size of the flying animal increases. For the 

 mosquito Aedes aegypti, weighing 1 milligram, Sotavalta's data show 

 0.2 percent of the total weight contained in the wings, whereas for 

 the falcon Gyps julvus, weighing over 7 kilograms, the wings, ac- 

 cording to Magnan, are 22 percent of total weight. 



WING-BEAT RATE AND WING LENGTH 



There is good evidence^ that the beating of the wings of flying ani- 

 mals can be described using the well-known theory for mechanical 

 oscillators. This theory presumes a resonance frequency for beating 

 wings which will be maintained regardless of changes in either external 

 or internal wing loading. It follows then that wing-beat rate will be 

 constant for a particular animal. The equation is as follows : 



,„ Khr'' 



^ =-T- 



where / is the wing-beat rate, br^ is proportional to the weight of the 

 wing muscles, and / is the moment of inertia of the oscillating system, 

 viz, the sum of the moment of inertia of the wings and the internal mo- 



^ Greenewalt, Crawford H., "The Wings of Insects and Birds as Mechanical 

 Oscillators," Proc. Amer. Philos. Soc, vol. 104, No. 6, 1960. 



