

On the undulatory Time of Passage of Light through aPrism.285 



origin is taken on the edge of the prism ; and let the position 

 of the initial point A be marked by three other rectangular 

 coordinates x\ i/, z*, having the same origin, but not necessarily 

 the same axes ; let a, /3, y, be the cosines of the angles which 

 the emergent or final direction CD makes with the rectangu- 

 lar axes of x, y, z ; and let a', /3', y 7 , be the cosines of the angles 

 which the incident or initial direction AB makes with the rect- 

 angular axes of ^,3/', z'; finally, let V be the undulatory time 

 of propagation from the initial to the final point, measured 

 by the equivalent path in vacuo ; and let it be considered as a 

 function of the initial and final coordinates, which, by the 

 position that we have assigned to the origin, is homogeneous 

 of the first dimension. We shall then have the two following 

 equations, deduced from my general methods, 



V = ax + Py + yz—a l j/—Pi/-'/&' 9 (1) 



o = xtct+ySp+zty-x'ta'-y' 8j3'-z'8y'; (2) 

 that is, V is to be determined as a function of the extreme co- 

 ordinates x y z x' y' s/ 9 which I have called in my Theory of 

 Systems of Rays the Characteristic Function^ by the condition 

 that it shall be the maximum or minimum, with respect to the 

 quantities a. f p, y 9 &', /3', y', of the expression (1): attending to 

 the two general relations, 



«* + /3 2 + y* = 1, a'* + £' a + y' 2 = 1, (3) 



and to the two other relations between the final and initial 

 cosines of direction a/3ya'/3'y', which result, in each parti- 

 cular case, from the prismatic connexion between the incident 

 and emergent directions. And when the form of the charac- 

 teristic function V is known, the six extreme cosines of direc- 

 tion may be deduced from it, by differentiation, as follows : 

 8V .8V 8V > 



*~ 17' <*- i*p V~JI> / 



1* "W *_ ii u m I 



a ~~ CO p ~ tyi 7 - izT ) 



When the prism is ordinary, such as glass, or when being- 

 extraordinary its edge is an axis of elasticity ; and when we 

 take the edge for the axis of z and of z\ and consider only 

 rays in a plane perpendicular to this edge, we niiiy make, 



z = o, z f = o, y = o, y' = o, \ . 



a = cos 0, /3 = sin 0, a! = cos 0', /3' = sin 0', J K °> 



being the emergent inclination to the axis of x, and 0' being 

 the incident inclination to the axis of ,r'; and the undulatory 

 time V, corresponding to any given coordinates xy x / 1/ 9 is 

 the maximum or minimum, relatively to 0, of the expression 

 V = x cos + y sin — x' cos 0' — ?/ sin 0', (6) 



