AIR-BUBBLES IN WATER. 



197 



Parallel rays, whose angles of incidence vary between 40 and 

 about 44, accordingly receive, by the internal reflexion combined 

 with the refraction at the surface, such a direction that they 

 all appear to come from points whose distance from the centre 

 amounts to '68 *7 . r, and therefore differs at most by -^ of the 

 radius. In other words, to an incident pencil of light composed 

 of parallel rays there corresponds an emergent cone of light, whose 

 virtual point of convergence falls somewhat below the plane 

 of adjustment, though almost on the line which represents the 

 ray emerging without deviation. At this point, therefore, is con- 

 centrated for the eye the whole effect of light which the incident 

 pencil is capable of producing in this way. In addition to this, 

 every other pencil of rays x y (Fig. 107), which with equal angles 

 of incidence deviates by a few degrees to the right or to the left 

 (drawn on the opposite side in order to avoid confusion), behaves, 

 with reference to the diameter-plane K L drawn at right angles 

 to it, in such a manner that the brightness of the point is 

 further increased by an infinite number of others, which lie 

 somewhat higher or lower, and form with it an uninterrupted 

 line of light. 



We thus explain the fact that a small space at a distance of 

 C = -69 . r from the centre, in the middle of the marginal 

 shadow, appears so luminous, and when viewed from above 

 appears as a bright ring. It is intelligible also that the breadth 

 of this ring, at any rate within the usual limits, increases and 

 decreases with the angles of aperture of the objective and of the 

 diaphragm, and that a higher or lower focal adjustment must 

 cause an obliteration of its outline. 



Applying the same reasoning to rays that have undergone two 

 internal reflexions, we can also explain the presence of a less 

 distinct outer ring. If S T (Fig. 108) is such a ray, which is 



