82 



and therefore, by (B), 



= COS. (X. ix -{- COS. ^1/, 3y -|- COS. ^z, iz -f- 



COS. ^'X. ix + COS. {'^. il/ + COS. {'«. Sz. (C) 



Now if we assume any point XYZ on the incident ray, at a distance { from the mirror, and 

 another point X' Y'Z' on the reflected ray at a distance 5' from the mirror, the distances of these 

 assumed points from the point x^dx, y-'r^y, «+3«, will be 



and because 



we shall have 



that is 



and finally, by (C) 



e = (X-*y- + ( Y-yfM Z-zf 

 e= {X'-xf + {Y'-yf-\-{Z'-zf, 



dj _ X—x d^ __ Y—y df _ Z—z 



dx i ' dy ~ ( ' dz~' j ' 



dj' _ X'—x dj' _ Y—y dj' Z—z 



dx {' ^ dy f' ' dz j' ' 



df df dg 



-^ = _ cos. «^> -^ = - COS. iy,-±=- COS. (X, 



—r- = — cos. fX) -T- = — cos. /y, — = — cos. e'z ; 

 dx dy '-^ dz * 



3j + V = 0. (D) 



This equation (D) is called the Principle of least Action, because it expresses that if the co- 

 ordinates of the point of incidence were to receive any infinitely small variations consistent with 

 the nature of the mirror, the bent path ({+?') would have its variation nothing ; and if light be a 

 material substance, moving with a velocity unaltered by reflection, this bent path 5 + 5' mea- 

 sures what in mechanics is called the /Action, from the one assumed point to the other. Laplace 

 has deduced the formula (D), together with analogous formulae for ordinary and extraordinary 

 refraction, by supposing light to consist of particles of matter, moving with certain determined 

 velocities, and subject only to forces which are insensible at sensible distances. The manner in 

 ■which I have deduced it, is independent of any hypothesis about the nature or the velocity of 

 light ; but I shall continue to call it, from analogy, the principle of least action. 



[4>.] The formula (D) expresses, that if we assume any two points, one on each ray, the sura 

 of the distances of these two assumed points from the point of incidence, is equal to the sum of 



