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 y 

 or the velocity be increased in the refracting medium, which agrees 

 with the molecular theory. 



On either supposition, if v = ?;/, and sin r positive, the case becomes 

 that of reflexion, and we have i = r, 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 i 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 I 



X O L M 



and R make equal angles with x x'. Then taking any other pairs of 

 lines i L, L R, and I M, 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 x' greater than i L x, and R M x' greater than i M x. Let I M and 

 L R intersect in K. 



Then we have the angle R L M greater than i L x, which is 

 greater than the opposite and interior i 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 i K=I 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 o; 

 therefore the lines meeting at o are a minimum. 



It is an extension of this principle which forms the basis of the in- 

 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. 1833. 

 Translator. 



