3 * 
Dr. Young’s Lecture 
Proposition VI. Problem. 
To determine the law by which the refraction at a spherical 
surface must vary, so as to collect parallel rays to a perfect 
focus. 
Solution. Let v be the versed sine to the radius 1 ; then, at 
each point without the axis, n remaining the same, m must 
become >/ mm^tiznv', and all the rays will be collected in 
the principal focus. 
Corollary. The same law will serve for a double convex lens, 
in the case of equidistant conjugate foci, substituting n for m. 
Proposition VII . Problem. 
To find the principal focus of a sphere, or lens, of which the 
internal parts are more dense than the external. 
Solution. In order that the focal distance may be finite, the 
density of a finite portion about the centre must be equable: 
call the radius of this portion f, that of the sphere being unity; 
let the whole refraction out of the surrounding medium into this 
central part, be asm tow; taker=. - 1 -— f — and let the den- 
sity be supposed to vary every where inversely as the power y 
of the distance from the centre: then the principal focal distance 
from the centre will be — . m — . When r = i, it becomes 
. — - — rn — -. For a lens, deduct one fourth of the difference 
2 (H. L. m — H. L. n) 5 
between its axis and the diameter of the sphere of which its 
surfaces are portions. 
Corollary. If the density be supposed to vary suddenly at the 
surface, m must express the difference of the refractions at the 
