196 



THEORY OF MICROSCOPIC OBSERVATION. 



will again employ the letter p. Since p is an exterior angle of the 

 triangle J B R, we get p = L B J R + L B R J, or, if a denotes 

 the angle of incidence, and a the angle of refraction, 



p = (a a) -}- a = 2 a a. 



It would be easy to determine from this equation the limits 

 between which the air-bubble appears to be illuminated in conse- 

 quence of the internal reflexion. The question, however, would 

 not even then be settled. It appears more to the purpose to take 

 a particular case, which leads directly to the solution of our 

 problem the case, namely, where the incident and emergent rays 

 lie in the same straight line, and consequently p becomes a right 



FIG. 107. 



angle. The difference 2 a' a attains the same magnitude ; from 

 which there results for a a value of 43J. If we now suppose, 

 instead of the incident ray, a parallel pencil of rays s t (Fig. 107) 

 to be taken, then only that ray whose a has exactly the stated 

 value proceeds in the original direction. All the other rays of the 

 pencil are deflected more or less ; and indeed, as calculation shows, 

 In the left of the perpendicular if a < 43 j, to the right if a > 43 J. 

 As examples a f ew callculated values are collected in the following 

 table. The first column contains the incident angle a ; the second 

 tlic corresponding deviation, </>, from the perpendicular to the 

 right or to the left ; and in the third, the distance of the points 

 in which the emergent rays intersect the plane of adjustment. 



