compound lenses and object-glasses. 245 
power of the compound being only about f of that of the 
first lens, is too low to be of service. 
On the same hypothesis as to the plane surfaces, if we 
trace the variation of the function -r, we shall find that 
0 + *) 3 ’ 
it admits a minimum for a positive value of x given by the 
equation 
37 a ? — 58 x — 68 = 0 
viz. x = 2*349. This gives 
L'= 2 35 x L ; L -f- U= 3*35 x L 
and, ~ = 0-24841 
This is the minimum value which the ratio of aberrations ad- 
mits for a positive value of x. The combination is repre- 
sented in PI. XIX. fig. 3 ; and we see that a very material 
superiority over the best single lens of the same power is the 
result of such a disposition, the aberration being reduced to 
less than a fourth part. Even if the plano-convex lenses 
thus laid together be of equal focus, the value of ~ will be 
only 0-6028, indicating still a sensible advantage gained over 
any single lens. 
13. Let us however take up the problem more generally, 
and enquire what should be the curvatures of all the four 
surfaces to destroy the whole aberration in the most advan- 
tageous manner with respect to the power of the resulting 
combination. To this end it is evident, that (the equation 
X=o still subsisting) we must also have L-|-L = a maxi- 
mum, and since we may assume as given the power of the 
first lens (without which the problem is indefinite), we have 
dL = o and dU~o, whence, dx = o also, differentiating then 
the equation X=o, on this hypothesis, we have 
