Note on Reduced Apertures. 109 



index of the medium, and the quotient will be the sine of the new 

 semi- aperture. 



Fir,. 1 . 



To exemplify this, let us take the glass experimentally tested in 

 the January number already referred to. In air the aperture was 

 found to be 145°, i. e. the semi-aperture = 72^°, and we wish to 

 find, without experiment, the aperture, e.g. for water. The sine 

 of 72^° is • 9537, which divided by f , the index of water, gives 

 0-7153, which number is the sine of 45° 40'. This, therefore, is 

 the semi-aperture for water, i.e. the aperture = 91°, agreeing with 

 the observed aperture within a fraction of a degree ; from which 

 we may remark, in passing, the extreme accuracy with which these 

 experiments must have been conducted. 



To find the aperture for balsam, we divide the same number by 

 1 • 549, the index for balsam, which gives for the semi-aperture 38°, 

 aperture 76°. The aperture observed was in this case 79°, which, 

 therefore, though within the maximum limit, appears to differ 

 slightly from what theory would indicate. This apparent discre- 

 pancy, however, is only half what it appears to read ; for it is to 

 be remembered that the angles ascertained by theory for comparison 

 are not the apertures, but the semi-apertures, and these in the present 

 case differ not by 3° but by \\°. In the commentary appended to 

 the record of the experiment, this difference is ascribed to the fact 

 of the balsam used having been very fluid — approaching turpentine. 

 To test this, the index of the balsam being in any case not so low 

 as pure turpentine, let us, at a venture, assume it to be an arith- 

 metic mean between the two. Computing with this index even 

 the slight difference recorded disappears, and the coincidence of the 



