250 The Theory of Imtnersion. 



sequent treatment at the different combinations is much the same 

 for all, 



A ray of light incident on the limiting surface of two media 

 is partly transmitted and partly reflected. And if the proportion 

 of the resulting intensity were always the same, our work would be 

 easy enough ; the effective light would in every case be given at 

 once by the versed sine of half the aperture. But the proportion 

 varies with every change, both in the nature of the media and the 

 obliquity of the incidence. The change in the media is definite and 

 easy, but the change in the obliquity advances by infinitesimal 

 degrees. So that to get at the total resulting amount we must 

 have recourse to systematic calculation. The results are easUy in- 

 telligible to everyone, and for purposes of experiment and testing 

 require nothing more difficult than to understand the meaning of 

 the word angle. But the computation itself is not of a kind which 

 the majority of the readers of this Journal would feel any interest in 

 following ; and from the general nature of the communications which 

 usually appear it would seem, I think, out of place to fill up the 

 pages with the details of arithmetical or mathematical work. Any- 

 one who wishes to test it can always fill in these details for himself. 

 But inasmuch as any value there may be in the investigation turns 

 entirely on its truth, I will first give in outhne the method which 

 I have followed, and for which I will ask only a single page ; so that 

 anyone so disposed may verify the work at his convenienco. 



A ray being incident at an obliquity 6 and refracted to an 

 obliquity 6', the intensity, I, of the transmitted hght is 



1 r sin.' (e + e ' ) - sin.'' (9 - e') ! 1 r tan.^ (e + 0') - ta n.' (6 - eQ -j ^ 

 "21 siu.2 ie + e') J "*" 2 L tan>'(e + 0') J ' 



one of the two terms giving the intensity of the light polarized 

 in the plane of incidence, and the other that polarized in the per- 

 pendicular plane. If we take the whole emitted Hght, corresponding 

 to a hemisphere, as our standard, or unity, then after loss by re- 

 flexion at the first surface, the transmitted light of the entire cone 

 corresponding to a semi-aperture 6 will be 



-± r P ''I sm.\e+e')-BmHe-e') ^ tan.'(e+e')-tan.'(e-e') ) ^^^ e^gd^ 

 4:irJ J y 8in.He+e') ' tan.^e+e') / ■ ^' 



in which 6' is, of course, an implicit function of 6 and the index 

 of the medium. One of the two integrations is effected instantly ; 

 the other, however, cannot unfortunately be got in any shape worth 

 having. To arrive at our second integral, therefore, we are obliged 

 to take the primitive and circuitous route of making it up by finite 

 summation. Let a semicircle be conceived drawn round the source 

 of light ; and let this be cut up into zones, as in the figure, by planes 



