136 On the Optical Advantages — [Monthy Mictoscana! 
beading is blacker. A broken line of beads is given with unusual 
sharpness as black as jet. The light playing in the focus of the 
minute refracting bead or lens, conforms more closely to the pheno- 
mena of a brilliant pomt—it swells and contracts with a change of 
focus—and its colours are transformed. We cannot yet see the 
black ring which undoubtedly surrounds the bead im nature, when 
viewed singly, but crescentric black shadows are formed en masse ; 
the halos of each spherule becoming commingled together, and the 
jet black shadows then start forth through the general haze into an 
exquisite tracery of shaded silicious beading. 
The results obtained in this paper would seem to strongly 
justify Hartnack’s observations. He asks, “Are large apertures 
an advantage to the microscopist or to the optician?” and de- 
clares “ the disadvantages are for the latter only.” It is clear that if 
by the aid of a water or other fluid film, a larger nascent pencil of 
rays can be transmitted to the eye by means of a smaller apertured 
objective, whose corrections are far less expensive and difficult than 
for the glass of enormous aperture, then the advantage of destroy- 
ing the excessive deviations caused by the aéro-refraction is self- 
evident.* 
To find the angle of total internal reflexion of a nascent ray 
originating from a brilliantly illuminated particle under different 
conditions of optical refraction, from a denser medium into a rarer, 
as from glass into air, water, or oil of turpentine, it is only neces- 
sary to take the largest angle, which causes the emergent ray to 
verge upon an angle of 90°, whose sine is unity. If, therefore, this 
angle be called the limiting angle of refraction, since 
Sin. ¢ = psin. ¢’ 
In the limit, 
Sin. ¢’ = . sin. ¢, when ¢ = 90°? 
1 
i 
Giving therefore » the values ascertained for refractions be- 
tween the given substances, g’ the limiting angle of refraction at 
which total internal reflexion takes place is readily ascertained, as 
shown in the Appendix by Logarithmic Tables. 
* Nobert’s lines are cut upon the under surface of the covering glass, and 
therefore the defining rays proceeding from a point in the groove may be con- 
sidered as immersed, or rather formed in the glass so as to produce the same effect 
nearly as a body mounted in balsam. On this account water lenses, or lenses 
immersed in other fluid (especially such as have a refractive index nearly equal to 
that of the covering glass), transmit extremely oblique nascent rays throughout 
more centrically than the dry. 
