METHODS OP PETROGRAPHIC-MICROSCOPIC RESEARCH. 
rays. From Fig. 13 it is evident that the same relation holds true for the 
points K and A"; the arc K'A' is accordingly brought to focus, point for 
point, in the image K A . If a small surface element alone be considered, 
A'C A'K' 
the relation 
AC " AK 
can 
A'C A'k' 
be written =@ (the 
AC Ak 
linear magnification) ; or from 
equation (u) 
A'C _n sin u _ 
AC n' sin u' 
This is the condition which 
must be fulfilled, if a small 
surface element normal to the 
axis at A is to be imaged by 
wide ray pencils, point for 
point. The points A and A' 
are called the aplanatic points of the refracting surface for the axis ACM, 
since they alone satisfy the above conditions. In the construction of oil- 
immersion objectives these two points are of great importance. 
(c) SPHERICAL ABERRATION IN SIMPLE LENSES. 
In Fig. 14 let L be a simple collective lens and P a luminous point on the 
FIG. 13. 
i i \. 14. 
principal axis PM. Let the concentric arcs about P represent the positions, 
after equal time intervals, of a wave impulse starting from P and expanding 
in all directions. On reaching the lens the speed of propagation of the wave 
is retarded and its wave-front assumes the shape represented in the figure. 
On emerging into air again, the original velocity is regained, but the wave- 
front is still further distorted into a complex warped surface whose radii of 
curvature (ray directions) do not converge to a single point on the axis, the 
axial part of the wave meeting the axis at P' while the marginal rays inter- 
cept the axis at P"c , a point much nearerthe lens than /*'. Thisdeviation of 
the emerging wave-front from the spherical form is called spherical aberra- 
