FURTHER REMARKS ON THE COMPOUND EVE. 573 
At first sight it may appear that the ray, after it becomes 
parallel to the axis, will remain parallel, but this is an error. 
One has only to consider the elementary wavelets of a dis- 
turbance to perceive that those nearest to the axis will be re- 
tarded most: we have before us the limit of geometrical 
optics. 
Karl Exner finds that when every layer increases in density 
in a definite function of its distance from the axis, and this 
function is parabolic, every ray proceeding from the focus /’ will 
fall upon the focus /.* 
The same result is arrived at by physical optics, since if 
f be a luminous point, and w w the wave front falling upon 
the cylinder, the convexity of the wave front will diminish 
until it arrives in the position 6 b, after which it will increase in 
a negative direction until it emerges as w’ w’, the focus of which 
is f. 
If the refractive cylinder is elongated (Pl. XL., Fig. 2, A, 
a, b, c, d), a series of foci will be formed at fff. Calculation 
shows that the distance F from c d is a recurring function of 
. where c is a constant, and / the length of the cylinder within 
certain limits, the limit being the distance between f and /. 
Neither Exner nor his brother appear to have considered the 
effect of a convex surface at c d on the emergent pencil. 
Suppose the length fe to be such as to give a negative value 
to the focus, the rays fg fg would be rendered parallel or con- 
vergent by the lens surface c h d. 
Pl. XL., Fig. 2,C is copied from Exner’s work [252]. It 
represents the inverted image, b a, in the crystalline cone, 
and the course of parallel rays from the points a b of a distant 
object ; these are brought to foci in the plane b a, and leave 
* It may be objected by some that it is extremely improbable that the 
density of the cone should increase towards the centre exactly as a parabolic 
function of the distance of each part from its surface, but if we suppose the 
cone to be deposited in successive layers, and to increase uniformly in 
diameter, it will not appear improbable that its density would vary inversely 
as the mass of material deposited in the unit of time ; and this is exactly the 
relation supposed by Exner in his calculations.—B. T. L. 
