AIR-BUBBLES IN WATER. 201 



Conversely, therefore, if the movement of the light takes place 

 from below upwards, the rays of the very oblique cone Jifg i must 

 unite to form the acute pencil a c p b, and hence illuminate 

 the point o. If h f and i g approach nearer to the perpendicular, 

 which would be equivalent to diminishing the diaphragm, then 

 a c and b p also approach each other. The pencil of light reach- 

 ing the objective is thereby weakened, but remains, afterwards 

 as before, inclined somewhat to the right. This inclination passes 

 into a contrary one if the air-bubbles are considerably more distant 

 from each other. The lines of direction d f and e g then approach 

 more to the horizontal; a c and b p are consequently removed 

 further to the left, as would occur if the air-bubble A revolved in 

 the same direction round its axis. The emergent pencil of light 

 consequently changes its direction to the axis of the Microscope in 

 accordance with the distance of the two air-bubbles. 



Similarly may be demonstrated the strengthening of the other 

 rings, and the intenser illumination of the margin. We have only 

 to construct the corresponding cone of light and to trace back- 

 wards single rays, as shown in Fig. 109, in order to account for the 

 different phenomena. 



If we consider the air-bubble as the refracting apparatus, 

 without reference to the plane of adjustment, it acts essentially as 

 a bi-concave lens. Its focal length, /, which is of course negative, 

 is determined by the formula 



/= r. 



2(- 1)' 



in which r is the radius, and n the refractive index of the 

 surrounding medium. Since the two principal points, as in every 

 sphere, coincide with the centre, the above expression is also equal 

 to the distance of the focal point from the centre. In oil with the 

 refractive index 1/5, / is equal to r, in water to approximately 

 f r, which values are reduced more or less (in water to about 

 *2 . r) through the aberration of the marginal rays. 1 



1 An incident cone of light is always refracted by an achromatic system of 

 lenses as if it were composed of rays of about the mean inclination, z'.e., it acts 

 as a conical shell of small aperture. Hence, it is evident that a cover-glass 

 upon the object produces an apparent approximation by a definite quantity, 

 although the mathematical expression for this approximation is dependent upon 

 the angle of inclination of the rays of light, and therefore yields no fixed value. 

 A cover-glass of 227 mic. in thickness causes, for instance, with Nos. 7 and 9 

 objectives of Beneche, an elevation of the object-point of 80 mic., whence the 



