RELATION OF APERTURE AND POWER IN THE MICROSCOPE. 253 
and consequently the focal length ( a 
LN] 
2*rmm. =1-12thin. 1°56mm.=1-16thin. 2°: mm. = 1-12thin. 
According to the views developed above, objectives of these 
short focal lengths cannot be dispensed with, under the present 
conditions of microscopical optics, for those lines of scientific work 
in which it is of importance to obtain the dest possible quality of 
the image (sharpness, definition), viz. such a degree of dioptrical 
perfection of the image as is not perceptibly inferior to the naked- 
eye vision of real objects, even on sensitive preparations. 
As has been previously observed, higher powers than are strictly 
necessary for exhausting the attainable apertures, are desirable, and 
even indispensable, for many particular purposes. These may be 
obtained satisfactorily by using higher super-amplifications with 
the same objectives. Inasmuch as in that case the aim is merely 
to enlarge the image without displaying mez details of the objects, 
the increase of the dissipation-effects with increasing v is not a 
serious drawback, because the visual angle of the minutest detail 
is increased in the same proportion. ‘Though the absolute pre- 
cision of the image will be diminished under the higher v, the 
relative will remain the same, and must still be sufficient for the 
more enlarged image if it was sufficient for the less enlarged. 
Now as ¢wice that super-amplification which raises the defects of 
the systems just up to the threshold of vision, is always borne 
by objectives without any considerable or objectionable loss of 
definition, the increase of the ocular-power alone will be sufficient 
for reaching the wer limit of generally useful amplifications for 
the various apertures which is shown by the fifth column of Table I. 
Nevertheless it may be desirable that such higher powers should 
be occasionally obtainable under still more favourable conditions, 
that is with the lowest figure of v. I admit, therefore, that 
objectives of shorter focal lengths—down to half the values 
assigned above at the utmost—may still be useful for the zmmer- 
ston-apertures. (Not for dry lenses, because it would be decidedly 
irrational to force such high amplification from apertures which 
leave the delineating power much below the attainable limit). 
On the other hand, it is quite certain that even the minimum 
powers which are required for the utilization of the said apertures, 
may be obtained under much higher super-amplification with a 
sufficiently satisfactory quality of the image, when the wtmost 
degree of perfection is not required or when objects of less sensi- 
bility, ¢.g., diatoms, are in question. Twice the critical value of v 
(z.e. 8, 8, 12, for the three systems respectively) will, however, be 
the limit in regard to objectives intended for somewhat general 
