26 
On the Limits of the Optical 
optical instrument than it was before, — a thing contrary to all 
experience, whatever kind of transparent refracting material be 
used. Now, if this were possible with light it would also be true of 
heat, as might be shown by application of similar reasoning ; and 
then the law of equal radiation of bodies possessing equal tempera- 
ture would be impugned. 
But the equation which premised very slight divergence angles 
of incident rays may be more precisely formulated, and so express 
the same result in the case of wide divergence angles. 
A more precise expression of the Law of Divergence Angles. — 
In equation [5 J it is a matter of indifference whether we substitute 
for a its sine or tangent or similar functions which for indefinitely 
small a would be its equivalent. If we assume larger divergence 
angles of a pencil of rays whose section is a circle, then 
fa 
L = J d S 2 TT . COS. a . sin. a . d a = irJ d8 . sin.^ a. 
J 0 
If after a series of refractions the surface d Si is completely 
n 
and accurately imaged in d Si with the brightness Ji and ai of 
the respectively appertaining divergence angles, then the amount 
of light must be 
n 2 
L = TT J . c? S, . sin.2 a, . 
As now, dS : dSi = P"^ : /3i^, there follows from these equations, 
w . /3 . sin. a = «i . i3i . sin. , [7] 
which renders this formula of equation [5] valid for larger angles 
of divergence, assuming that and y6i are two images exactly re- 
producing each other, and whose surfaces are perpendicular to the 
axis. 
Brightness of Image. — When the pupil of the observer's eye is 
fully immersed in the pencil of rays proceeding from any point of 
an image, the observer will see the image illuminated as brightly 
as the object. This result was already announced by La Grange. 
Unfortunately he had not investigated a second case, which happens 
to be more common just when high powers are used, namely, when 
the pencil of rays does not entirely occupy the pupil of the eye. 
If a pencil of light having only small divergence angle a, does 
not entirely fill the pupil when the image fii is situate at the proper 
distance of distinct vision, then the brightness H of the retinal 
image in that eye will be less than that entering the free eye Ho, 
whose pupil is entirely filled with light. 
