
PROFESSOR MARTIN DUNCAN’S ADDRESS, 277 

incident pencils entirely by their angles, has been its inevitable 
tendency to give a fictitious importance to the angle of the entering 
pencil, which was supposed to have a special virtue of itself, in the 
delineation of objects. Naturally, therefore, the same angles, 
whether in air or any immersion fluid, were considered to produce 
an equal effect, and the advantage of immersion objectives was 
rested on minor points. 
An estimation of the emergent beam, however, must obviously 
give the same result as one of the incident beam (assuming them 
both to be correctly made), it being of course impossible for any- 
thing to emerge that has not first been admitted. But to quote 
Mr. Crisp :—“ The great and obvious advantage in dealing with 
the emergent pencil is that it is always in air, and so the perplexities 
are eliminated which have enveloped the consideration of the 
admitted pencil, which may be in air, water, oil, or other substances 
of various refractive indices.”* 
The subject of aperture is not, in reality, a difficult one, and any 
intricacy in which it may seem to be involved will be found to arise 
from the necessity of clearing away some of the old entanglements, 
such as the curious mistake involved in the “hemisphere puzzle” 
and similar matters. Looked at de novo, there are two simple 
Stages in the aperture question. 
(1) To appreciate that, in using the term “aperture” we use it 
not in any artificial sense, but as meaning opening and nothing 
else,—defining, simply, the capacity of an objective for receiving 
rays from the object and transmitting them to the image. 
(2) That the aperture (as so defined) of an objective is determined 
by the ratio between the diameter of the emergent beam and the 
focal length of the objective. According as this ratio is greater or 
less, so the objective will receive and transmit a larger or smaller 
portion of the total quantity of rays presented to it. 
The emergent beam of an air objective of 180° angle cannot 
exceed in diameter twice the focal length ; that of a similar water- 
immersion objective may be one-third larger, and of an oil-immer- 
sion half as large again, and the relative capacities of such objectives 
(with equal angles) to receive and transmit rays will always be as 
1, 1% and 1. 
It cannot be too carefully borne in mind that it is not a question 
® As pointed out by Mr. J. Mayall, jun., at the commencement of the discus- 
sion, if 180° in air is equivalent to 82° in glass, the 140° in glass of the immersion 
lens must represent something more. ‘This fact is, , So constantly mis- 
interpreted, owing to the supposition that when the immersion fluid is introduced 
onl is no longer compressed by the action of the plane 
surface of the lens, but is allowed to expand to 140°. is is one only of the 
a difficulties the proper estimation of the incident pencil, 
are avoided by dealing with the emergent beam. 

