PRINCIPLE OF LEAST ACTION. 191 
duced it from the principle of attraction, because that law 
which can only agree with observation provided v/ be greater than »v, 
or the velocity be increased in the refracting medium, which agrees 
with the molecular theory. 
On either supposition, if v= v/, and sin r positive, the case becomes 
that of reflexion, and we have i=7r, which is the law of reflexion, 
whence Ptolemy’s conclusion is manifest as a particular case of the 
general theory. The case of reflexion is, in fact, nothing more than a 
geometrical problem. 
Let two points 1 R, be given without a given straight line x x/, and 
let o be the point in that line at which straight lines drawn from 1 
BY 
BR 
K 
a; 
_s 
x 6 b&b M x/ 
and rR make equal angles with x x’. Then taking any other pairs of 
lines I'L, L R, and 1M, M R, terminating in the same points and meet- 
ing x x/ in L and in M, they will each form unequal angles with x x/; 
R L xX! greater than 1 L X,and R M x! greater thanim x. Let1m and 
L R intersect in K. 
| Then we have the angle R & M greater than 1 L x, which is 
greater than the opposite and interior 1 m L; and therefore in the 
triangle K L M, K M is greater than K L. 
In the limit, when M approaches L, we have ultimately 1 K=1 L, 
and K R=M R; whence I L+L K+K R is less than I K+K M+M R, or 
the pair of lines nearest to o are together less than the more remote. 
The same reasoning will apply to all pairs of lines on either side of 0; 
therefore the lines meeting at o are a minimum. 
It is an extension of this principle which forms the basis of the in- 
i vestigations of Sir W. R. Hamilton. Observing that in some parallel 
instances the action is, in fact, not a case of minimum, but of max- 
imum, he has adopted the more generic term, “stationary action; ”’ 
. and upon this has based his fundamental idea of the “ characteristic 
. function,’ by the aid of which his profound analytical system, ap- 
. plicable equally in questions of optics and dynamics, is constructed. 
. For an admirable exposition of the general principle the student 
should consult Sir W. R. Hamilton’s paper on “ The Paths of Light 
. and of the Planets”? in the Dublin University Review, Oct, 1883.— 
4 Translator. 
