28 METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
time required for the wave disturbance to travel from P to P' must be the 
same for all radial directions emerging from P; the same holds true for the 
conjugate points A and A 1 . Accordingly, the optical distances 
PNEP' = PGFP' and AKEA' = AHFA' 
therefore 
AKEA'-PNEP'=AHFA'-PGFP' = AHF+FA'-PGF-FP' 
but a plane wave-front PA becomes a spherical wave, after refraction, with 
F as center; therefore PGF = AH1<; in like manner the plane wave-front 
DP becomes spherical after refraction with E as center, and DKE = PNE; 
as the distance PA is indefinitely small we may consider as a first approxi- 
mation the length EP' = ED', in which case A KEA ' = PNEP' = DKED' and 
therefore D'A ' = DA , an equation which referred to vacuum becomes 
n'.P'A'.sm (-u') = n.PA sin u 
But 
PA 
accordingly 
n' sin u' _ i , v 
n sin M /3 
or 
sin u' _ n i , . 
sin M w'/3 
This formula states that the ratio of the sine of the axial angles u and u' 
of any two conjugate points in plane through the aplanatic points normal 
to the axis is constant. If the Abbe sine condition be properly met, the 
chief and marginal rays from the point A will converge to the point A', as 
illustrated in Fig. 19; but in actual practice the point A represents any 
point in the field, and \vc have; therefore, under these conditions, correction 
of spherical aberration in the plane passing through the object point and 
the optical axis (tangential or meridional plane}. If the sine condition be 
fulfilled, the images of the object produced by the central and marginal 
zones are identical in size and position. But the sine condition may also 
show zones, just as spherical aberration proper; if the marginal zone fulfills 
the sine condition, the intermediate zones do not generally do so. Systems 
that are spherically corrected and fulfill the sine condition are, by Abbe's 
definition, called aplanatic. Aplanatism is possible only for one pair of 
conjugate points. 
In the actual construction of microscope objectives, these correction scan 
be effected only for three zones simultaneously (central, marginal, and an 
intermediate zone); these are then so chosen that the deviations of the 
remaining zones are small and exert only slight influence on the resulting 
image. In immersion objectives the aplanatic points are of prime impor- 
tance, as the front lens of such systems is simply a small unconnected glass 
hemisphere. The object is placed in the near aplanatic point and the 
emergent wave is strictly spherical and of lower aperture than the original 
pencil. In dry objectives such ideal conditions arc not possible. 
