THE GAUSS EQUATIONS FOR PARAXIAL RAYS. 
_^ _ e__ . ,_ e 
y\ /.-yV ' "ft- 
In similar manner it may be proved that d = 
&-A 
These equations prove that the distance d' is independent of y, and the 
distance d, of y'; in other words, all rays incident parallel with the axis pass 
FIG. 6. 
through F' t the focal point of the second medium, while all rays emerging 
parallel with the axis pass through F, the focal point of the first medium. 
From Fig. 6 it is evident that 
h h 
or 
==/' (6) 
tan ' fa /3j 
where/' is the principal focal length in the second medium. In like manner 
we may prove that 
" =^%=/ (6.) 
in which / is the principal focal length in the first medium. Equations (6) 
and (6a) define the focal length of an optical system as the ratio of the 
height of an incident ray parallel with the axis to the tangent of the angle 
which it includes with the axis on emergence. 
If in the first medium or object space the measurements be referred to F 
as origin of coordinates, and in the second medium or to image space F' as 
origin of coordinates, the positive direction in each coordinate system being 
from left to right and above the axis, then from Fig. 5 we have 
BF h B'F' h' 
m'= =777= 7 (7) 
(8) 
On substituting these values in (6) and (6a), we obtain 
**'=//' 
