AIR-BUBBLES IN WATER. 195 



This latter difference occurs for the assumed index of refraction, 

 if a = 43; therefore a' =65 37'. The distance of the limiting 

 pointy, in Fig. 105, is given by the formula 



sm 60 



Under a Microscope with the assumed angle of aperture, the air- 

 bubble will therefore exhibit a black band, the breadth of which 

 = (1 - -7875) r = '2125 . r. The central part up to P would 

 appear just as bright as the field of view, and the zone between 

 P and p would obstruct the light. An alteration in the angle 

 of aperture, by the employment of different objectives and 

 diaphragms, would only modify the numerical relations, but not 

 the general distribution of the light. With greater values for o> 

 and S the black band would become narrower, and with smaller 

 values broader. Oblique illumination would produce an excentric 

 position of the bright circle. 1 



In reality the microscopic image of the air-bubble presents an 

 essentially different appearance. The bright inner part appears 

 encircled by a dark zone, which gradually merges into a perfectly 

 black one ; this zone does not extend to the margin, but is encircled 

 by a bright ring, and again by a somewhat uniform penumbra to 

 the periphery. With more careful observation we recognize, a 

 little to the outside of this bright ring, a second one less distinct ; 

 indeed, the whole penumbra proves to be made up of alternately 

 bright and dark concentric circles. 



These phenomena cannot be explained by refraction alone ; 

 they are due to rays, which have in addition undergone one or 

 two internal reflexions. Let T L (Fig. 106) be such a ray, which is 

 reflected at R and refracted a second time in J"; then, as is evident 

 from the figure, J B C is the half angle of deviation, for which we 



1 Harting (" Das Mikroskop," 2nd ed. ii. pp. 30 and 32, note) does not agree 

 with this theory, according to which the angle of aperture o> has no influence 

 upon the breadth of the dark zone. He relies upon his observations and 

 measurements, according to which the diameter of the bright central part is 

 simply and solely determined by the angular magnitude of the source of 

 light, dependent upon the aperture of the diaphragm. That is, however, 

 erroneous, and we cannot understand how he could obtain such a result. If 

 we place a piece of paper or tin-foil, with a minute perforation (a needle-hole), 

 as a diaphragm between the objective-lenses, and then observe the image of a 

 spherical air-bubble, it will not be necessary to resort to measurement in order 

 to establish the influence of the angle of aperture. 



