GAUSS’S DIOPTRIC RESEARCHES. 491 
theorems already mentioned, the thickness of the lens may be 
taken into account without any loss of simplicity ; the only limit- 
ation retained being that the inclination of the rays to the axis 
is very small, or that the spherical aberration is neglected. 
[It is worthy of observation, that in most of the elementary 
treatises on optics in use at present in this country, the point in 
the lens from which the focal length is measured is distinctly 
stated. Also those points in a single lens which are called 
“haupt-puncte ” by M. Gauss, have been long known to En- 
glish optical writers under the name of “ focal centres.” (Wood’s 
‘Optics,’ fifth edition, Cambridge, 1828, Art. 165. Coddington 
*On Reflexion and Refraction, Art. 69.) An investigation of the 
principal relations between the distances of the conjugate foci of 
a thick lens measured from the focal centres will be found in 
Wood’s ‘ Optics, Arts. 194. 469. 470.—Ep.] 
2. To determine the path of a ray after refraction through any 
number of media separated by spherical surfaces having their 
centres in the same straight line. 
Let RS T be the path of a ray through two media bounded 
_ by spherical surfaces meeting the line through their centres in 
| A,B, RS and therefore ST being in the plane ABS. Let 
RS, ST meet the axis A B in P, Q;; then p, »! being the indices 
_ of refraction of the media in which RS, S T lie, v the radius of 
the surface BS, supposing the inclination of the ray to the axis 
to be very small, ultimately 
SIs tine Bae 
OB 6 PB ee 
whence, observing that 
SB RA  ,SB RA 
ee fe wl = pe —f 
mop, “ont at 7 8B 
also 
RA 
SB=RA+ PA AB. 
When BS is a reflecting surface we have only to put w! = — p. 
By this substitution the whole of the following investigation may 
be extended to the case in which one or more reflexions occur. 
3. Let A B be the nth of a series of s + 1 media separated by 
2K 2 
