398 PROFESSOR KELLAND ON FRESNEL'S FORMULAE FOR THE 



those of the same particle at the time t, 



x+8x, 7/ + 8i/, z + 8 z, and z + Ss + Sy 

 the corresponding quantities for another particle in the upper medium ; 



x + dx,, ^j + 8i/,, z + 8z„ z + Sz^+dy^ 



the co-ordinates of another particle in the lower medium at the same time. 



When discussing the lower medium separately, we will adopt a;,,t/,, z^, y^ etc. 

 in all cases for which we use x, y, z, 7, etc. in the upper. 



r is the distance between the two particles in a state of rest. 



r + g their distance in a state of motion. 



Let rfprhe the force on the particle under consideration, arising from another 

 particle at the distance 7\ If, however, it be thought requisite, we may consider 

 r (p r as the aggregate attraction of a group of particles about a material particle. 

 The law of force, whenever a law is wanted, will be assumed to be that of 

 Newton. 



Let 7^, 8 X' etc. denote the distance, etc. of particles in one medium from those 

 in the other. 



The notation 7/^ -s^ „j.s„ denotes the value which y, assumes when 

 x,+8x^, y^ + 8yj are written for x^ and y^. 



Slight deviations from these arrangements will occasionally be made, which 

 will be pointed out when they occur. 



SECTION I. 

 ON LIGHT, CONSISTING OF VIBRATIONS PERPENDICULAR TO THE PLANE OF INCIDENCE. 



We shall adopt the following process : first, deduce the equations of motion 

 on the supposition that the force is insensible except at very small distances from 

 its origin, and then take the law of force, varying inversely as the square of the 

 distance. The object of the first process is the discovery of the form which the 

 results assume, to serve as a guide to the more complex calculations of the 

 second. 



The two media will be supposed to be arranged in a perfectly symmetrical 

 manner, so that all terms which involve the odd powers of the distances of the 

 particles, will vanish when the sums of such terms are taken, extending through- 

 out the whole mass. 



1. The expression for the force on the particle P, resolved parallel to the 

 axis of ;:;, is 2 (^ {r + §) {8z-v8y). 



= 1 {(p r + (p' r . g) (8z + 8y) 



= l((pr+^Ji 8z8y) {8z + 8y) 



= 2{(pr.8z + i-L8z^8y + (pr.8y] 



