SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
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aberrations in such a form that, in a combination of surfaces and lenses, the effect of 
a given surface or lens on the final result can be readily traced. This is fundamental 
for the designer, who usually proceeds to sketch out his system by Gaussian methods 
only, being guided therein by considerations of magnification, illumination, and field 
of view; and then goes on to eliminate the resulting image defects, so far as lie can, 
by bending the lenses, i.e., by altering their mean curvature without changing the 
focal length. In doing this he usually corrects one defect at a time, with the frequent 
result that, when, having corrected one defect by means of one lens, he proceeds to 
correct a second defect, he thereby causes the reappearance of the first. 
If the effects of any given lens, however, are made apparent in the final formula, it 
becomes a more manageable problem to devise variations which will keep any one 
defect invariant whilst others are being dealt with. 
§ 2. Notation. 
There is no general agreement among mathematical writers as to the notation 
employed in dealing with optical problems, and it will be convenient to state here 
the symbols we have adopted. They are a modification of a system due to 
Steinheil. 
The successive media, proceeding in the direction of travel of the light (from left to 
right in our figures), are denoted by even suffixes 0, 2, 4, &c., and the same suffixes 
affect the rays in these media, their inclinations, a 0 , a 2 , a 4 , &c., to the axis, and their 
intersections I 0 , I 2 , I 4 , &c., with that axis. 
The successive geometrical images will be denoted by the letter J, thus J 0 , J 2 , J 4 . &c. 
The successive surfaces of separation will be denoted by the odd suffixes 1, 3, 5, &c., 
and the same suffixes will affect the centres of curvature, the intersections of rays 
with the surfaces, and the points where the axis crosses the surfaces. The latter 
will be denoted by the letter A and the centres of curvature by the letter C. 
Fig. 1 illustrates the use of this notation for two refracting surfaces. 
The radii of curvature are r 4 , r 3 , r 5 , &c., and are to be considered positive when 
A 2n+4 C 2n+ i is measured from left to right. 
The perpendicular from a centre of curvature on a ray is denoted by p and is 
affected by a double suffix, the first belonging to the centre of the curvature and the 
second to the ray. Thus p v2 is the perpendicular from the centre of curvature (\ of 
the first reflecting surface upon the ray in the second medium. Where there is no 
ambiguity the first suffix will usually be omitted. 
The refractive index will be denoted by n and affected by the suffix of its 
medium. 
Transverse magnifications will be denoted by M. The magnification produced by 
surface 1 will be denoted, as convenient, by Mj or M 02 ; by surfaces 1, 3 combined 
either by M 13 or M 04 : by surfaces 1, 3, 5, combined either by M 135 , or M 0 „, and so on. 
F 2 
