SPHERICAL ABERRATIONS. 27 
image, the point A appears drawn out like the tail of a comet, which maybe 
directed away from or toward the margin of the field. This aberration is 
usually called coma, the fuzziness being caused by the difference in magnifi- 
cation by the different zones of the objective. The first approximation for 
the removal of coma is the fulfillment of the Abbe sine condition, which 
states that the ratio of the sines of slope angles between any two incident 
FIG. i 8. 
rays and the two resulting emerging rays is constant (in other words, the 
optical paths for the different rays from object point to image points are 
equal). The condition that all rays from the axial point P unite in P' 
(Fig. 19) postulates that for these conjugate points the system is free from 
spherical aberration, while the condition that all rays from A unite in A' 
requires that the system be free from spherical aberration for the secondary 
axis A MA'. The mathematical expression for this condition has been 
deduced in a number of different ways, the simplest being possibly that of 
Hockin.* In Fig. 20, let PA and P'A' be small conjugate central surface 
elements ; from P and A let two parallel rays emerge which intersect at E 
FIG. 20. 
in the image space ; let also F be the point of intersection in the image space 
of the axis with the ray through A parallel to the axis ; let DP be normal to 
the incident ray A and D'P' to the refracted ray KA '. If the spherical wave 
disturbance emerging from P is to be focussed at P', then P' must be the 
center of curvature of the spherical wave surface in the image space, and the 
*Jour. Roy. Micros. Soc. (a). 4, 337. 1884. 
