ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 357 



length of the light (of any definite colour) in air, and by 8 the 

 distance of consecutive elements of the structure, then the wave- 

 length of the same rays within a medium of the refractive index n 



will always be -. By the Fraunhofer formula the angles iPj, 



W2 ... of the first, second . . . deflected beam within the 

 medium are defined by the equations 



\ \ 



Sin ii'i = - ; sin w^ = 2 - > 

 



or 



Sin w. = —r ; sin v:„ = 2 — - 



{n is of course = 1 for air). 



In passing from air to the medium n the sine of the deflection 

 angle of the first, second . . . bent-off beam is reduced, therefore, in 

 the exact proportion of n, and the angle itself is reduced also — that 

 is, the whole fan of the difiracted rays is contracted in comparison 

 with its extension in air. 



If now any dry objective (with a given angular semi-aperture 

 «) is capable of gathering-in from air the first, or the first and second 

 diffraction beams on every side of the axial pencil, another objective 

 with a more dense front medium of the refractive index n, will be 

 capable of admitting, from within the dense medium, exactly the same 

 beams (no more and no less), if its angular semi-aperture v is less 

 than u in the proportion — 



sin V '. sin u = 1 ' n, 



or if 



n sin V = sin u, 



all other circumstances — object and illumination — remaining the same. 

 For example, a diatom for which the distance of the strise is • 6 /x, 

 will give the first bent-off beam of green light {X = ' 55 /x), in air 

 under an angle of 66-5°. This will be just admitted by a dry 

 objective of 133° angular aperture. In balsam (n = !• 5), the same 

 pencil will be deflected by 37*5° only, and would be admitted 

 therefore by an objective of not more than 75° balsam-angle. Again, 

 if the distance of the lines should be greater, as 1*2 /x, the second 

 deflected beam would be emitted in air under an angle of 66*5°, but 

 in balsam the third would attain the same obliquity. Whilst now the 

 dry objective of 133° air-angle cannot admit more than the two first 

 diffraction beams on each side of the axis, the immersion of 133° 

 balsam-angle is capable of admitting from balsam three on each side 

 under exactly the same illumination. 



It follows, therefore, that a balsam-angle of 75° denotes the same 

 aperture as an air-angle of 133°, and a balsam-angle of 133° a much 

 greater aperture than an air-angle of the same number of degrees, and 

 in general two apertures of different objectives must be equal if the 



