256 THE MICROSCOPICAL NEWS. 
the apertures 0°85—-0°g0, we have obtained a numerical determina- 
tion of ¢iree points of the series of apertures in the dry system ; 
and we shall arrive at an approximate estimation for the inter- 
mediate points by interpolating the values of v between the said 
three points. The annexed tabular statement exhibits in the fourth 
column the series of v which results from the data given above, if 
the interpolation is made by means of a parabolic curve. The third 
column repeats the figures of the minimum total amplification N 
required for the various apertures, according to Table 1.; the fifth 
column shows the objective-amplification [N] which results from 
the corresponding value of v; and the sixth column gives the 
focal length f which will afford that objective-power in every single 
case. Retaining fractions in the values of v and f, whilst the basis 
of the calculation is not determinate, has of course no other pur- 
pose than to prevent arbitrary leaps in the series of figures, which, 
according to the nature of the case, must show a continuous 
gradation. * 
According to the observations from which the figures of v have 
been derived, ¢77p/e systems of moderate working distance must be 
supposed down to apertures of about 0°3, and duplex systems for 
all the lower ones which are considered here. The medium aper- 
tures, if realized with two lenses only, and the lower ones, if 
realized with single lenses, would show considerably smaller figures 
of vy than are given above. On the other hand, triple systems with 
apertures much below 0°3 do not afford any perceptible increase 
of the admissible super-amplification—a fact which is well accounted 
for by the theory of the aberrations. 
So far as the basis of my reasoning is admitted as valid, the 
table given above will exhibit the proper ratio of aperture to focal 
length in an zdeal series of dry objectives of increasing apertures, 
traced out in strict conformity to the principle that every objective 
should yield, under the dest posstble conditions, such a total amplifi- 
cation as is just sufficient for fully exhausting the delineating power 
of its aperture—wherein “ best possible conditions ” means that no 
higher super-amplification, by tube and ocular combined, should be 
required than that which will jws¢ raise the dioptrical defects of the 
image up to the threshold of vision. 
It is not my opinion that the standard series thus obtained 
should always be strictly adhered to in the practical construction of 
objectives. It is rather advanced here as a theoretical guide which 
will give a general direction in designing systems on a rational 
basis, but does not prohibit any deviation from that standard, 
provided it be justified by this or that practical consideration. 
Deviations in the direction of diminished aperture (or increased 
power) need not be discussed here, since at present no tendency of 
that kind is met with, except in Microscopes of quite an inferior 
