The Theory of Immersion. 



251 



perpendicular to the axis, each zone being thus the part of the 

 hemisphere intercepted between two consecutive right cones whose 

 common axis is the axis of the object-glass. From one of these 

 cones to the other the proportion of light reflected will of course 



vary. Let the mean value be taken, and then if the zone be cut 

 thin enough we may without any appreciable error assume the 

 value of the multiplying coefficient which gives the lost light (or 

 the transmitted light) to be constant throughout the zone. In 

 this way, by adding all the zones together we get the whole 

 amount of effective light corresponding to any original semi-aperture 

 6. At the commencement or near the axis the change is extremely 

 slow ; the loss at 20° e. g. being very little different from the loss 

 at 0°. But as the angle increases, the change in the coefficient 

 becomes very rapid, and for all the higher apertures the belts must 

 be taken very thin, or an appreciable error would arise. 



Taking the sum of these up to any angle we wish to know, 

 we get the whole effective light for that angle, and adding in the 

 intermediate ones we get the value for the next aperture that we 

 wish to record. This must be repeated for all the bounding sur- 

 faces. In the case of a dry object we have to reckon with two 

 losses at the surfaces of the cover, and another at the front itself. 

 Using an immersion front, we get likewise three diminutions — 

 one at the under surface of the cover, with the same index as before, 

 and two more with a difl:erent index, i. e. from the cover into the 

 water, and the water into the front. 



Taking, first, the case of an object not in any medium, the 

 results are given in the subjoined table for as many apertures as 

 appear to be of use for the present purpose of comparison. 



VOL. XI. u 



