compound lenses and object-glasses. 
2 35 
or, when the last refraction is made into a vacuum, 
/= F-D (/"') 
5. Let us imagine a system of n lenses , each consisting of 
a single medium, placed close together in vacuo, and as the 
ray after traversing each separate lens emerges into a va- 
cuum, we have tx = = . . . . &c. = 1 ; m m = 1, m m =1, 
&c. and therefore k = u, — 1 = — k ,k = «. 1 = — k. &c. 
1 * 1 2 3 1 3 4 
so that our expression for F becomes 
F=i, (r,— r 2 ) + A, {r— r + ) + &c. 
or, denoting jt/y &c. simply by p! , &c. 
F= (f» — 1) (r— : '■,) + (!«.'— 1) (r— r + ) + &c. (g) 
In the case of a single lens, this reduces itself to its first 
term, and calling L, L', &c. the powers of the several lenses, 
we have 
L= 0 - 1) ( r,-r 2 ) ; L'= O'— l ) f r } -r + ) ; &c. (A) 
and finally 
F= L + L'+ L"+ &c. (z) 
which expresses that the power of a system of lenses ( placed 
close together and infinitely thin), is the sum of the powers of 
its component lenses. The powers of concave lenses are here 
regarded as negative, as well as their focal lengths ; while 
the equation (J'"j shows that the sum of the reciprocal dis- 
tances of the object and its image is equal to the power, or 
reciprocal focal length, of the system. 
6. These propositions are sufficiently well known, and 
comprise the whole theory of the central foci of infinitely 
MDCCCXXI. H h 
