MAYEK: COLOR AND COLOR-PATTERNS. 193 



being hard, chitinous, and inflexible, would serve but poorly as a 

 tactile or sensory surface. 



Of course no one would venture to ascribe any sensory function to 

 the scales which cover the wing-raembranes of the Lepidoptera. 

 We may, however, make several more or less reasonable hypotheses 

 concerning the probable uses of the scales, and by testing these sup- 

 positions arrive pei'haps at some plausible explanation of then- reten- 

 tion and the complex development which they have undergone. 



(1) They may have caused the wings of the ancestors of the 

 Lepidoptera to become more perfect as organs of flight, by causing 

 the frictional resistance between the air and the wing-surface to 

 become more nearly an optimum. 



(2) The appearance and development of the scales may have 

 served, as Kellogg ('94) has suggested, "to protect and to strengthen 

 the wing-membranes." 



(3) The present development of the scales may be due to the 

 fact that they displayed colors which were in various ways advan- 

 tageous to the insects. 



Concerning the first of these three hypotheses, the wing has, 

 broadly speaking, two chief functions to perform in flight. It must 

 beat more or less downward against the air, and must, in addition, 

 glide or cut through the air, supporting the insect in its flight. For 

 the mere beating against the air a relatively large co-eflicient of 

 friction between the air and the wing might be advantageous ; but 

 for gliding and cutting through the air a 5/na// co-efiicient of friction 

 would certainly be an advantage. There must therefore be an 

 optimum co-efficient of friction, which lies somewhere between these 

 two. 



In order to determine the co-eflicient of friction between the wing 

 and the air, use was made of a method which, in one form or 

 another, has long been known to engineers; that is, of observing the 

 ratio of damping of the vibrations of a pendulum. 



It is well known that when a pendulum is swinging free, and 

 uninfluenced by any frictional resistances, the law of its motion is 

 expressed by the formula, 



■277 



(i) d = A sin ^ t 



where d is the displacement of the pendulum from its middle 

 position after the interval of time t, A is the maximum displace- 

 ment and T the time of a complete vibration, back and forth. If, 



