AIR-BUBBLES IN WATER. 193 



If we now suppose a second adjacent ray Z K, which is less 

 inclined to the perpendicular and which also appears after refraction 

 to come from the point P, it is evident that at some point it 

 intersects the first ray, and therefore meets the objective somewhat 

 nearer the centre. For if we follow the two rays in the opposite 

 direction, from above downwards, their point of convergence, P, 

 moves slightly to the left, in consequence of the refraction at the 

 surface in J, assumed to be infinitely small, and hence lies in a 

 different level ; an intersection takes place invariably within the 

 air-bubble. This applies also to each succeeding ray with 

 reference to the preceding one. The further we proceed in the 

 emergent cone of light from left to right, the further is the 

 corresponding incident ray moved from right to left, its in- 

 clination to the perpendicular becomes gradually less, and then 

 passes- into the opposite inclination, whose maximum also amounts 

 to 15. 



It is also evident that this maximum deviation must occur 

 before the emergent rays on the right have reached the limit of 

 30, as the two refractions always cause a deviation to the left. 

 Calculation shows that in the given case the marginal ray T'L is 

 inclined, after its passage through the air-bubble, slightly more 

 than 18 to the left. 



These discussions lead first to the conclusion, that all rays of 

 the incident cone of light 1 lying in the plane of the paper con- 

 tribute to the illumination of the point P. Moreover, of the rays 

 not lying in this plane none are lost. For since, after refraction, 

 they all appear to proceed from the point P, and furthermore lie 

 in the same plane with the radius drawn to the point of emergence, 

 consequently cutting the plane of the paper in the line M N, we 

 shall exhaust all possible positions thereof, if, while retaining 

 the point of convergence, all rays between S J and S'J 7 (Fig. 104) 

 are raised above the plane of the paper, and allowed to diverge so 

 far upwards and downwards, that the corresponding incident rays 

 graze the margin of the diaphragm. That the inclination to the 

 plane of the paper of the rays thus raised can at the most amount 

 to 15, in the given case, and for the two marginal rays =0, is 

 evident without further remark. There is, moreover, no difficulty 



1 The expression cone of light is not to he taken with strict mathematical 

 accuracy for the incident rays, inasmuch as they have no common point of 

 convergence. 







