ME. E. A. SAMPSON: A NEW TEEATMENT OF OPTICAL ABEEEATIONS. 1G7 
The part depending upon 6? represents the spherical aberration; the part 
depending upon /3^, if it were present alone, would indicate that the images, if we 
can so call them, were found upon a sphere of curvature —K(4G + 4H); the 
Fig. 5. 
Ot, m-)» 
corresponding expressions for the primary focal line in place of the focal circle would 
be — K(4G + 24H) and for the secondary focal line — K^;jG. These expressions are 
positive when the sphere is convex to the incident rays. 
The angular values of the radii of the focal cii'cle and the comatic circle are 
respectively 
( 1 ) ( 2 ) 
206265'" X. ri/3V/' and 206265" x 
Thus with increasing aperture {d) the focal radius increases with the first power 
and the comatic radius with the square, while with increasing breadth of field (,8) the 
focal radius increases with the square and the comatic radius with the first power. 
To fix ideas we may consider the case of the parabolic reflector; here, as shown 
on p. 156, 
diG = 0, 4G = +1/2/', S,,G = 0, SJl = +1/2/', = -1, = 0. 
Hence spherical aberration and distortion are absent. ^G = 0 implies that the 
secondary focal line lies in the normal focal plane, while the focal circle lies upon a 
surface of curvature l/2/'; and for the efiects of astigmatism and coma we have the 
following table for difierent apertures and fields* :— 
* Cf. Poor, ‘ Asti’ophysical Journal,’ VII. (1898), p. 121. 
