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 « = «/, and sin r positive, the case becomes 

 that of rejlexion, 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 e, 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 



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

 lines I L, L E, and i m, m k, terminating in the same points and meet- 

 ing X X' in L and in m, they will each form unequal angles with x x' ; 

 E L x' greater than i l x, and r m x' greater than i m x. Let i m and 

 L E intersect in k. 



Then we have the angle k 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 k; whence i l+l k+k r is less than i k+k m-|-m k, 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 wliicli 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, lie 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 tlie student 

 should consult Sir W. E. Hamilton's paper on " The Paths of Light 

 and of the Planets" in the Dublin University Review, Oct. 1833. — 

 Translaiar. 



