194 Proceedings of Royal Society of Edinhurgh. [sess. 
For axial rays these general values become respectively 
■ {-) 
■ (^) 
The difference between these two values of the principal focal 
distances (measured from the opposite surface) is evidently 
2R 
fX-l 
(For IX — 1-5 the difference is 4R.) 
The principal points (as can be shown to be the case in other 
ways) must therefore lie to the side of the convex surface at 
It 
distances respectively of from each surface and separated by 
fx-l 
a distance = t. 
If the principal focus be measured from the principal points, we 
It 
have to add or subtract from the above values. 
fX-l 
It then becomes in both cases 
which shows that the focus of such a lens increases directly as the 
square of the radius of either surface^ and inversely as the thichness. 
(y) may be written 
1 fxR R 
t fX -I ' IX 
by which its relation to the formula for refraction at one surface is 
shown. 
/ 6R2\ 
(For ju,= 1-5 y becomes F = ) * 
The difference between (a) and (/?), or, in other words, the 
position of the principal points, accounts for the different degree of 
neutralisation produced according as one side or other of a com- 
bined convex and concave lens of equal focus is held to the eye. 
