498 GAUSS’S DIOPTRIC RESEARCHES. 
e=a— +, fab+4, fon=-(-nst 
(f—9) (p—¢) = =e 
12. The angles subtended at M, N by the object and its image 
respectively are equal. 
linear mag. image _ v _m—e_ f—q 
The image will be erect or inverted according as the numerators 
and denominators have the same or different signs. 
13. In a combination of two lenses ¢, $, being the focal lengths; 
m, ” the distances of the first and second focal centres of the first 
lens from the fixed point O; m!, n! the distances of those of the 
second lens; m, n the distances of the first and second focal 
centres of the combination, 
1 ] rx a 
Ries t,=n—ml, e= Si a—b=m—n', m=m, n=N, 
n—m! + n—m'+o+¢9! n—m' + >} 
og =e, h=n—m', ia gare = ae 
ay MOM) O49 pn (Om Oa ee 
i nO Saal n—m'+o+ 9!” 
sts —_m! x => 1 1 
orgie (n—m') > ea. (n—m!') 
(x — m')? 
Hence, when m —m! is small compared with either ¢ or $', as is 
the case in an achromatic object-glass, the distance between the 
centres of the combination is very nearly equal to the sum of 
the distances between the focal centres of the separate lenses. 
It is evident that all the formulz of the present article may 
be applied without any change to the case in which instead of 
simple lenses partial systems of lenses are combined into one 
system. 
— n—m+o+o" 
~m—n=m—n+ m'— ni + 
