414 
Proceedings of the Royal Society 
incidence and the axis, these angles being the respective diver- 
gences, we have rigorously by the law of refraction 
sin (y - a) = p sin (y - /3) , 
or, approximately , 
y -a = p(y-(3), 
or pP - a = (p - ,l)y . . . . (1), 
where p is the refractive index. [This we may, if we choose, 
translate into 
where y is the distance of the point of incidence from the axis, and 
the rest of the notation is as usual. In this form we see that, to 
our approximation, the result is independent of y .] 
In (1) we have y=0 for a plane surface, and p = - 1 when there 
is reflection instead of refraction. 
Hence for a reflecting surface the meaning of (1) is — u the sum 
of the divergences of the incident and reflected rays is twice that 
of the normals to the surface.” If the incident rays be parallel, 
the reflected rays diverge twice as much as do the normals. 
At the second surface of a thin lens (1) becomes 
which, compounded with (1), gives 
P' - a = (p - 1 ) (y - y') , 
which may be thus translated — “ A lens produces a definite change 
of divergence on any direct pencil — and the change is p - 1 times 
the difference of the divergences of the normals to its surfaces.” 
Hence that a divergence may be changed into an equal negative 
divergence, it must be equal to half the change produced by the 
lens; i.e ., when the object and image are equidistant from the 
lens, their common distance from it is double the focal length of 
the lens. 
