AIR-BUBBLES IN WATER. 



199 



mately the twenty -seventh part of the half - diameter ; it appears 

 most sharply outlined at a somewhat higher level. Its lower 

 intensity of light is intelligible, partly by the loss caused by the 

 two reflexions, partly by the narrow limits of the angles of 

 incidence (about 45 J 47). 



We may hence conclude that further bright rings are formed by 

 pencils of rays which have undergone internal reflexion three, four, 

 or more times. It will be sufficient for these cases to collect the 

 conditions under which the emergent rays show no deviation, and 

 thence determine the points in which (if produced backwards) 

 they cut the plane of adjustment. The distances of these points 

 may be identified with those of the corresponding rings without 

 further explanation. 



It is evident that this series terminates as soon as the angle of 

 incidence has reached its limiting value, which amounts for water 

 and air to 48 29'. The ring corresponding to this limiting ray, 

 which, however, is no longer perceptible, would be '7487 . r from 

 the centre, that is, approximately, f of the half-diameter. Never- 

 theless, the remaining portion of the margin may still be illuminated 

 by the internal reflexion. From the table we might rather conclude 

 that all the rays here under consideration, which do not contribute 

 to the formation of the rings, are deflected so that they appear to 

 come from points which lie without the corresponding ring. A 

 feeble illumination of the margin must therefore result, while the 

 umbra appears, within the rings, perfectly black. 



Since the distances of the rings are dependent upon quantities 

 which are determined by the index of refraction of the surround- 

 ing medium, we might infer that liquids of greater density (as 

 oil, glycerine, &c.) would modify the numerical relations above 

 given. In oil of refractive index 1-5 we obtain for the innermost 

 bright ring a distance of 5957.7* from the centre, which agrees 

 perfectly with observation. Hence the rings always lie somewhat 



