compound lenses and object-glasses . 241 
ix = it ” & c. = 1; m m ™ 1 ; m m ==, 1 . &c. 
^2 r 4 1 2 ’34 
But we have also 
/ 3 =^{v,+ y 2 +y 3 - D } 
=^{y3+/ 2 } 
whence we see that is formed from —f 2 precisely ( mu- 
tatis mutandis ) as f is from D ; and it is therefore evident 
that if we take L # , A', B', C', the same functions of the re- 
fractive index and radii of the second lens, that L, A,B,C, are 
of those of the first; and put D'= — / = — (L — D) and 
write f / for p, , the refractive index of the second lens, we 
must have 
V-, Q 3 + ^ 4 Q 4 =| x f-' 1 l'{a'+ B'D'+ CD*} 
and similarly for the value of p,.. Q -{- Qs adding another 
accent to the letters in the second member, and observing 
that jLt /r = p, and 
D"== - (L' — DO = — (L + L'— D) 
and so on, so that we have ultimately, whatever be the num- 
ber of lenses, 
A/= | { m.*L(A+BD+CD-)+/'L'(A'+B'D'+C'D'*)+&c. J . (o) 
continued to as many terms as there are lenses. 
If the surfaces of a compound lens be in optical contact, 
( "1 . e. if the media of which it consists, instead of having thin 
lenses of air or vacuum interposed, be contiguous, the con- 
vexity of one fitting exactly into the concavity of the other, 
as in the case of two glass lenses inclosing a fluid), we may 
still regard them as separated by infinitely thin, non-refrac- 
tive laminae, having equal curvatures on both sides, for it is 
obvious that these will produce no deviation. In this case, 
