. Biographies of distinguished scientific men. Scientists; genealogy. PRINCIPLE OF LEAST ACTION. 191 duced it from the principle of attraction, because that law â which can only agi-ee with observation provided v' be greater than r, or the velocity be increased in the refracting medium, which agrees with the molecular theory. On either supposition, \iv = vi, 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 t
. Biographies of distinguished scientific men. Scientists; genealogy. PRINCIPLE OF LEAST ACTION. 191 duced it from the principle of attraction, because that law â which can only agi-ee with observation provided v' be greater than r, or the velocity be increased in the refracting medium, which agrees with the molecular theory. On either supposition, \iv = vi, 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 K, 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 -K' 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 xMn 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 >r greater than i l x, which is greater than the opposite and interior i >i 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-j-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 paii's of lines on either side of o; therefore the Hues 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- plica
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