WHICH REFLECT LIGHT, 30? 



number of spheres of corresponding size, by the calculus of 

 probabilities. Fixing our attention, in the first place, upon the 

 forward surface where the rays enter, in the case of arbitrary 

 bodies we must, according to the laws of probability, assume 

 that surfaces of all possible directions occur in the same manner. 

 This is the case with each sphere in particular ; for it is the 

 essential characteristic of the sphere that its curvature is every- 

 where the same, and thus the elements of the surface which cor- 

 respond to the various directions have all the same magnitude. 

 In regard to the hinder surface a difference however exists. In 

 the case of arbitrary bodies, the direction of one surface is inde- 

 pendent of that of the other, and purely accidental. But in the 

 sphere, the hinder surface, where it is met by the ray, has the 

 same inclination towards the latter as the forward surface, inas- 

 much as every cord of the sphere meets the spherical surface 

 under two equal angles. The question now is, what influence 

 will this peculiarity of the sphere have upon the total reflexion 

 and the total dispersion. 



In the case of reflexion this influence is not important. If 

 the whole quantity of light reflected at the forward surface in 

 the case of the sphere be equal to that reflected by the arbitrary 

 body, the same may be assumed of the second reflexion at the 

 hinder surface, and in like manner of the third, fourth, &c.; 

 interior reflexions respectively, at least in so far as the dimi- 

 nution which the ray has suffered by the previous reflexions, is 

 not taken into account. As all these quantities of reflected 

 light simply add themselves together, it follows that the total 

 quantity of light which is reflected from a great number of ar- 

 bitrary bodies, is nearly equal to that reflected from the same 

 number of spheres of corresponding magnitude. 



It is otherwise with the refraction. Here the essential cir- 

 cumstance enters, that in the case of a sphere the second re- 

 fraction takes place in exactly the same direction as the first, 

 while with arbitrary bodies the second plane of refraction can 

 have any inclination whatever towards the first. The difference 

 however can be definitely expressed numerically. Let us sup- 

 pose the two single deflections small enough to permit of the 

 angle being set for its sine, it is easy to deduce that the average 

 value of the double deflection in the case of spheres is to that in 



