ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 
255 
however, given at the end of this paper have not, so far as I am aware, 
been published before. 
In order that all the points may be perfectly clear, a few lines of 
elementary optics are necessary. It is obvious that the rays pass twice 
through the lens and undergo one reflection from the mirror. If / be 
the focus of the lens, and /' that of the mirror, F the focus of the com- 
bination, when its thickness is neglected, will be 
! = ? + !. 
p f f 
(0 
Now if fx is the refractive index, r the radius of the incident 
surface, and 8 that of the silvered surface, / the focus of the lens taken 
alone will be 7 and f that of the mirror ~ • 
O - i) (« 7 r ) . . . 2 
Putting these values in equation (i.) we have in terms of the radii 
the focus of the entire system. 
When fx = - 
A 
F = 
vs 
2 { M* — »•) — «}■ 
(ii.) 
(Hi.) 
Therefore, if n = 
and the lens is equiconvex s = — r, then 
r = 4F^ but if the lens is a plano-convex, r being the plane side = 00 , 
and s = 3 F, that is the radius of the silvered side is four and three 
times the focus respectively. 
With regard to the aberration of lens mirrors, it is obvious that in 
the equiconvex form an amount of refraction equal to that of an equi- 
convex lens is obtained with only a fourth of the depth of curvature. 
In the plano-convex form the incident parallel rays pass through 
the plane surface without aberration, but then there is the spherical 
aberration of the concave mirror, as well as the aberration on the final 
emergence of the rays at the plane surface ; in addition to this, the 
radius of the curvature of the mirror is less in the proportion of 3 to 4 ; 
nevertheless, calculation shows that the aberration of the plano-convex 
is less than that of the equiconvex form. For example, when y is the 
g 
semi-aperture, F the focus, and n = - , the aberration of an equiconvex 
32*4 y 2 7/ 2 
lens mirror is — ^ — , or • 5~ , that of a plano-convex mirror lens being 
8 • 5 t / 2 y 2 
— , or *315 ; whereas the aberration of an ordinary plano-convex 
