compound lenses and object-glasses. 
«3S 
diminished. In this case the refractive power of the medium 
varies by insensible gradations ; and if we suppose both it 
and the radii of curvature of the layers of equal density to 
vary according to a given law, we shall have both (x and r, 
expressed in functions of the depth to which the ray has 
penetrated at any moment of its course. This is the case 
with the crystalline lens of the eye. Dr. Brewster’s ob- 
servations have demonstrated that this humour increases 
very rapidly in density from the circumference to the centre ; 
and to apply our general equation to the evaluation of its 
focal length, we must proceed as follows. Taking x to re- 
present the depth of any layer whose thickness in the middle 
is da ?, the curvatures r and r' of its surfaces will be r and 
r + dr. We have also, t~dx, t'—dx-\-d 2 x, taking x for the 
independent variable. 
Moreover, since — t = m ! , we have 
t* 
?n = 
t*+dp 
d/x. 
d[x. 
and in consequence, 1 — m!=. -A Hence we obtain 
q>'— ( 1 — m 1 ) r'— 
neglecting the products of the infinitely small quantities, so 
that our equation (c) becomes (since/' =f -^-df) 
f + d f= ~f- + — 
which developed, retaining only terms of the first degree, 
gives 
/+#=/+/* *-/*+ 
rdm 
or simply, putting p = e* where e = 27182818, &c. 
o — 4f+ (/—'") do—pdx. 
00 
