8 Prof. Kirchhoff on the Relation between the Radiating and 



plane. Lastly, at the distance of unity from the last plane, and 

 parallel to it, let a fourth plane be taken, containing a system of 

 rectangular coordinates parallel to those in the third plane, and 

 having the axis of the pencil for its origin. Let x 4) y 4 be the co- 

 ordinates of any point in this plane. 



From the point x x y x a ray is supposed to proceed to the point 

 xflq. Let the time it takes to pass from one point to the other 

 be called T. Then T is some function of x x y v #„y 2 , which we will 

 suppose to be known. If the points x 3 y 3 , x 4 y 4 lie in the path of 

 the ray, and if for the sake of brevity the velocity of the ray in 

 vacuo be taken as unity, then the time required to pass from 

 x 9 y 3 to x 4 y 4 will be 



=T- Vn-^-^ + y,-^ 2 



- Vl+a? 2 -ff 4 V + y 2 -y 4 l 8 - 

 If the points x^y^ x 4 y 4 were given, and the points x x y lt x 2 y 2 

 were required, they might be found from the condition that the 

 above expression is a minimum. Supposing, therefore, that the 

 eight coordinates, x v y v x 2 , y 2 , x 3 , y 3 , x 4 , y 4i are very small, the 

 condition that the four points of which they are the coordinates 

 shall all lie in the path of the same ray is expressed by the fol- 



Now let x x y x be a point in the projection of surface 1 on the 

 plane x x y u and let dx x dy x be the element of this projection which 

 contains the point x x y Xi and which must be considered as infi- 

 nitely small as compared with the surfaces 1 and 2. Let x 3 y 3 

 be a point in a ray which proceeds from 1 to 2, dx 3 dy 3 the su- 

 perficial element containing the point x 3 y 3> and of the same order 

 of magnitude as dx x dy x . The intensity of the rays whose waves 

 are of the length already mentioned, and which are polarized in 

 the given plane, and proceed from x x y x through # 3 yg, is then, 

 according to § 5, 



d\\ dx x dy } dx 3 dy 3 . 



Now according to the hypothesis we have assumed, the pencil 

 arrives at 2 with its intensity undiminished, and forms an ele- 

 ment of the magnitude indicated by Kd\. Whence for K we 

 must have the integral 



taken between the proper limits. 



The integration according to x 3 and y 3 must be between the 



