HOLLOW SPHERES AND HOLLOW CYLINDERS. 



211 



which for the inner bounding surface of the wall are the angles of 

 incidence and refraction, we will denote by a" and a"', and the half- 

 angle of deviation C P T by p. Then p = L K T - (a- a), or, 

 since C K T is an exterior angle to the triangle U K J 



(nr //\ /// o /// // 



a a ) -\- a = z> a a , 



p = 2 a" -{- a (a" -\- a), 



. .# . , R . U 



wnere sin a = sin a = sin a. and sin a = n sin a = 

 T nr T 



sin a, n being the relative index of refraction of the substance of the 

 cylinder. As the ray emerges without deviation we obtain the condi- 

 tion 2 a"' + a (a" -f a) = 90. If the surrounding as well as the 

 enclosed medium is water, and the refractive index of the hollow 



cylinder n = we obtain for the following ratios between r 



I'ooOD 



and R the values of a subjoined, and thence the distances F of the 

 bright line from the centre. 



r 1 



In the second series = was assumed = , by which 

 H n 



sin a" = sin a, and therefore a" = a. 



The comparison of the last column with the first shows that the 

 values of F are somewhat smaller than those of r, or, in other 

 words, that the inner bright line falls in the hollow space. It is 

 evident that its distance from the wall will increase and decrease 

 with the refractive index ; for since a" is not dependent upon n 

 then, if we bring the equation of condition to the form 



2 a" - [a + (a" -a)] = 90, 



the expression within the square brackets will be smallerwh en n, 

 increases, because in that case a" and a, and at the same time 

 their differences, become smaller also. We must therefore assume 

 a larger in this case to satisfy the equation ; and the bright line 

 is moved further inwards. Similarly may be explained the move- 

 ment outwards of the line when n decreases. These displacements 



