Capacity of the Mieroseope. By Prof. Helmholtz. 23 
Again, if we consider the short arc h Ji of the refracting surface 
as a straight line at right angles to the axis a h, 
hh' = ah . tan. a = — bh . tan. a". 
Or substituting the angles for the tangents, which is allowable 
here on account of the smallness of the angle, 
^ = ^- [2] 
OL ah 
Multiplying equations [1] and [2], we get 
a . ^' _ac .hh 
a" . ^" ~ be . ah ^' ^ 
Now, according to the known laws of refraction at a spherical 
surface, whose radius he = the value of their principal focus is 
. n' r ^„ , n" r 
• ¥' = hf=- F" = hg = ~ [4] 
from which follow 
F' _ n' 
F" - F' = r. [4''] 
Further, 
or 
F' F" , , F" F' , 
— + — = 1, and — + — = 1 
a h 0 h a c 0 G 
hh bh -F" , he be -F' 
and 
ah F' ' ac F" 
Division of the last two equations gives 
bh,ac _ F" (b h - F") _ 
ah .bo ~ F' (6'c - F') ' 
but by equation [4^], 
bh = bc + r = bc +F" -F', 
and 
6A-F" = 6c-F'. 
Hence 
bh.ac F"- n" ^ ... 
— ; — — = — = — , according to equation r-i*]. 
ah . be F' n'' ° ^ ^ 
Therefore equation [3], 
or 
n' .0.'.$'- n" . a" , jS". [5] 
q. e. d. 
