356 
Proceedings of the Roycd Society 
to angles of elevation of the piece, respectively less and greater than 
45°. In the former case (when the elevation is less than 45°) a 
slight increase of elevation increases the range on a horizontal plane, 
so that the new path is wholly above the old one. In the latter 
case a slight increase of elevation shortens the range, so that the 
two paths must intersect. 
It is then shown that the equation of the curve of vertices is 
where the refractive index is 
i x = Jf(y) 
(the axis of y being vertical), and b is the vertical distance of the 
eye from the axis of x. 
The author first tried the case of an arrangement of strata with one 
stratum of minimum refractive index. As the ordinary change of a 
quantity near its minimum value is proportional to the square of its 
distance from the minimum, the assumption made was 
/x = Ja 2 + y 2 , 
without any inquiry into the dynamical stability of such an arrange- 
ment in air. 
Here, of course, 
f= y^iog .» + Vg E? . 
rj 
This curve is easily traced by drawing, separately, the rectangular 
hyperbola 
£= fa 2 + y 2 , 
and the curve (whose ordinates are the reciprocals of those of a 
catenary) 
c. -i b 4 - 
£ = log — 
V 
or 
2b 
V = .. . 
e s + e £ 
and multiplying together the values of $ for each value of rj. 
It was then seen that whenever b is greater than 3.68a, this 
\lb 2 — rj 2 
