1888.] Lord McLaren on an Aplanatic Ohjective. 
375 
(% + (^2 ~ “(^1 “ ^^3 
= = =|-1 ( 21 )- 
Writing, in the left-hand part of the above expressions, 6^ for 
(n^ -t- we have 
/^A + K- = 
/^l^S "h /^i^2^2 ~ f*'1^2 “ 2^3 = 0 
^ _ (/^l ~ ~ /^1^2 _ (/^l 7 AA _ 2 _ A ~ _ 2 
/^1^2 /^1^2 /AjA 
Substituting for A and A' their values, this expression becomes 
^ + 1 _ *^2 A “ ^) _ /// “ 2)(/^2 ~ 1)(^2 “ 0 
Whence is obtained the reduced value of 
^ _ A - 2)(/^2 - 1)/// + / 90 \ 
(;x,-2)(/X2-1K-/x,;x2F 
It is noticeable that the value of does not depend in any degree 
on the second surface, but solely on A', w^hich is the ratio of the 
focal length of the first surface to that of the instrument. is 
therefore determined entirely by the condition of achromatism. 
That being so, the compound lens will be aplanatic, if the second 
surface be any curve of the form, /(sec n^O^), a form which includes 
the circle. Its two arbitrary constants, n and a, are determined 
by the condition of minimum deviation in the two lenses. As this 
need not be very accurate, we see that the figure of the intermediate 
surface is of less importance than that of the first surface, on which 
accordingly the skill of the optician should be concentrated. 
The quantities here found being computed, these are to be 
introduced along with the values of a-^a^ into the respective equations 
/(cos /(sec n^B). 
The computation of a standard table of ordinates is then extremely 
simple. Making a= 1, we find 
log = log cos log rg = log 7^2 sec ^2 • • • (2^)* 
A series of values of r-fi^ r^B^ being thus obtained, we find corre- 
sponding values for and the ordinates (l-a?^) (l-a; 2 ), to 
which any required multipliers can be applied, . . (24). 
A table of ordinates applicable to the mean densities of crown 
and flint glass given may be used without sensible error for testing 
