THIS GAUSS EQUATIONS FOR PARAXIAL RAYS. 15 
as a first approximation PM for pM and P'M' for p'M 1 ; in other words, a 
small surface element normal to the axis is pictured, by refraction at the 
spherical surface, point for point, as a small surface element normal to the 
axis. The planes PM and P'M' are called conjugate planes. 
From Fig. 3 and equation (i) we obtain 
P'M' ^CM' y'^x'-r^n a/ 
PM CM ' r y x-r n'' x 
if PM = y and P'M'= -y'. 
This equation states that the lateral magnification 
y n' x 
is constant for any pair of conjugate planes and only varies from pair to pair. 
FIG. 3. 
Similarly, from Fig. 2 we find with the same degree of approximation 
the angular magnification 
. tan u' x 
On combining (4) and (3), we have 
= * (5) 
which states that the product of the lateral and the angular magnifications 
is constant. In equation (5) neither x nor x' appears. 
REFRACTION THROUGH A LENS. 
In the above paragraphs we have considered the refraction at a single 
spherical surface S, and found that an image P'M' is produced of a lumi- 
nous object PM (Fig. 3). This image is similar to the original object and 
may serve in turn as a luminous object for a second refracting surface Sj 
(Fig. 4), which produces an image P"M", similar both to P'M ' and PM. 
Strictly speaking, P'M' is not a luminous object similar in every respect to 
PM, but one from which only a limited cone of light is emitted, not sufficient 
to fill the entire aperture of Sz, as is the case with the luminous object PM 
and the refracting surface Si. For a series of two or more centered refract- 
ing surfaces, therefore, as in actual lenses, the image M' from the first 
surface becomes the object or axial point for the second surface, and is pic- 
*The Smith-Helmholz or Lagrange-Helmholz equation; see J. P. C. Southall. Geometrical Optics. 368. 
1910. 
