358 
Proceedings of Royal Society of EdinlurgJi. [june 4 , 
and 2). The equation of the aplanatic surface for infinite rays is 
easily found. There are two cases. In figure 1 the rays are 
supposed to he passing in a state of parallelism through glass, and 
to converge to a focus S in air. In figure 2 the rays are supposed 
parallel in air, and are brought to a focus in glass by refraction at 
the surface PQ. 
The axis of the lens is the axis of X, and the optical focus S 
is the origin of coordinates, ir, y, r, 0. As the direction in which 
the course of the rays is traced is immaterial, I shall here consider 
the course of the rays as diverging from S, and becoming parallel by 
refraction at the aplanatic surface. 
SPI' SQP are two consecutive rays diverging from S, and refracted 
at PQ into parallelism to the axis. 
P?^ is drawn perpendicular to SQ, produced if necessary. 
Pr is drawn perpendicular to IQ, produced if necessary. 
is the refractive index from air to glass in the first figure, and 
from glass to air in the second figure. 
are the angles of incidence and refraction at the point P. 
Then in the elementary triangles, QP^, QPr, — 
8r = 8SP = ?zQ = PQ . sin QP?? = PQ sin 
hx = SIP = rQ = ± PQ . sin QPr = PQ sin 
Sr sind) . 
Hence, by integration, r - fx^x - C = 0 in the first figure : 
r + (XqX - C = 0, in the second figure, (2). 
In each of the figures jx^ is the refractive index from the first 
medium to the second medium. Hence fx^ in the second figure is 
the reciprocal of in the first figure. 
If, instead of fx^ we use /x, the refractive index from air to glass, 
we have for the case in the first figure 
Sr = fxSx; r = /xx + Cj (3); 
and for the case in the second figure, — 
8r = — Sir; r = i a? + C (4). 
fx fx ' ^ 
The curve is a conic section, as may be seen by squaring and 
Avriting x^ + y^ for r^. 
To determine jx and C we observe that by a known property of 
