MAYER: COLOR AND COLOR-PATTERNS. 193 
being hard, chitinous, and inflexible, would serve but poorly as a 
taetile or sensory surface. 
Of course no one would venture to ascribe any sensory function to 
the seales which cover the wing-membranes of the Lepidoptera. 
We may, however, make several more or less reasonable hypotheses 
concerning the probable uses of the seales, and by testing these sup- 
positions arrive perhaps at some plausible explanation of their 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 frietional 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 (9+) 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. 
Coneerning 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-efficient of 
frietion between the air and the wing might be advantageous; but 
for gliding and cutting through the air a small eo-eflieient of friction 
would certainly be an advantage. There must therefore be an 
optimum co-eflicient of friction, which lies somewhere between these 
two. 
In order to determine the co-efficient 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, 
. 2m 
(dis Asin Tt 
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, 
