within the principal focus, they will neither be 
brought to converge nor be rendered parallel, 
but will diverge in a diminished degree. The 
same principles apply equally to a_plano- 
convex lens, the distance of its principal focus 
being understood to be the diameter of the 
sphere. They also apply to a lens whose sur- 
faces have different curvatures; the principal 
focus of such a lens is found by multiplying the 
radius of one surface by the radius of the other, 
and dividing this product by half the sum of 
the same radii. For the rules by which the 
foci of convex lenses may be found for rays of 
different degrees of convergence and divergence, 
we must refer to works on optics. 
The influence of concave lenses will evidently 
Fig. 150. 
a 
Parallel rays falling on a plano-concave lens made to 
diverge as from its principal focus, and rays con- 
verging to that focus rendered parallel, 
be precisely the converse of that of convex. 
Rays which fall upon them in a parailel direc- 
tion will be made to diverge as if from the 
principal focus, which is here called the nega- 
tive focus. This will be, for a plano-concave 
Fig. 151. 
i 
a 
Ni! 
‘i 
WHA = 
— 
==35, 
———_— 
— 
= == 
= = 
=F 
E ; 
Parallel rays made to diverge as from the principal 
focus, and rays converging to that focus rendered 
parallel. 
Fig. 152. 
Rays greatly converging made to converge less, and 
rays slightly diverging made to diverge more. 
lens, at the distance of the diameter of the sphere 
_ of curvature; and for a double concave, in the 
centre of that sphere. In the same manner, 
Tays which are converging to such a degree that, 
MICROSCOPE. 
333 
Rays slightly converging made to diverge. 
if uninterrupted, they would have met in the 
principal focus, will be rendered parallel; if 
converging more, they will still meet, but ata 
greater distance ; and if converging less, they 
will diverge as from a negative focus at a greater 
distance than that for parallel rays. If already 
diverging, they will diverge still more, as from 
a negative focus nearer than the principal focus; 
but this will approach the principal focus, in 
proportion as the distance of the point of di- 
vergence is such, that the direction of the rays 
onions the parallel. 
f a lens be convex on one side and concave 
on the other, forming what is called a meniscus, 
its effect will depend upon the proportion be- 
tween the two curvatures. If they are equal, as 
in a watch-glass, no perceptible effect will be 
produced; if the convex curvature be the 
greater, the effect will be that of a less powerful 
convex lens; and if the concave curvature be 
the more considerable, it will be that of a less 
powerful concave lens. The focus of conver- 
gence for parallel rays in the first case, and of 
divergence in the second, may be found by 
dividing the product of the two radii by half 
their difference. 
Hitherto we have considered only the effects 
of lenses upon a pencil of rays issuing from a 
single luminous point, and that point situated 
in the line of its axis. If the point be situated 
above the line of its axis, the focus will be 
below it, and vice versi. The surface of every 
luminous body may be regarded as comprehend- 
ing an infinite number of such points, from 
every one of which a pencil of rays proceeds, 
and is refracted according to the laws already 
specified ; so that a perfect but inverted image 
or spies of the object is formed upon any 
surface placed in the focus, and adapted to re- 
ceive the rays. 
In optical diagrams it is usual, in order to 
avoid confusion, to mark out the course of the 
rays proceeding from two or three only of such 
points. By an inspection of the subjoined 
figures, it will be evident that, if the object be 
placed at twice the distance of the es 
focus, the image being formed at an equal dis- 
