SPHERICAL ABERRATIONS. 21 
SPHERICAL ABERRATIONS. 
MONOCHROMATIC LIGHT. 
The assumptions made in the above paragraphs on the Gauss or first 
order equations, that only paraxial rays (the narrow bundle of rays immedi- 
ately adjacent to the axis) are to be used, is not applicable to the lens system 
of the microscope. Here it is necessary for a number of reasons to employ 
lenses of larger opening. From each point in the object a pencil of rays of 
wide angle emerges and of this the objective should collect as much as 
possible and bring it to sharp focus at a point in the image if the definition 
is to be satisfactory. This condition should hold for all points of the object. 
The image, in short, should be similar in every respect to the object and as 
bright as possible. In actual lens designing and lens construction there are 
practical difficulties in the way of fulfilling this condition accurately in all 
details, and the lenses which the observer receives are more or less encum- 
bered with defects or aberrations (deviations from the theoretically perfect) 
which it is impossible to eliminate entirely. If monochromatic light be 
used, these aberrations may be conveniently grouped under five heads: 
(i) Spherical aberration proper. (2) Spherical aberration of oblique rays 
(sine condition). (3) Astigmatism. (4) Curvature of field (Petzval con- 
dition). (5) Distortion (tangent condition). 
SPHERICAL ABERRATION PROPER (FOR THE PRINCIPAL AXIS), 
(a) REFRACTION AT A SINGLE SPHERICAL SURFACE. 
In Fig. 10 let HA be a spherical glass surface (refractive index = 1.50) and 
P a luminous point on the axis PA . A light-wave impulse starting from P 
travels with equal velocity in all directions and the shape of the wave-front 
at any given instant is that of a spherical shell concentric to P. The spheri- 
cal arcs in Fig. 10 represent the different positions which a part of the wave 
impulse sent out by P reaches after equal successive time intervals. On 
entering the glass the wave travels more slowly and the wave-front is no 
longer a spherical surface, but a warped surface, the normals (ray directions) 
of which are represented by the arrows of Fig. 10. Optically, it takes the 
wave impulse the same time to travel from b B (Fig. 10) in air that it does 
to travel from A to A' in the glass (the ratio of b B to A A ' being by defini- 
tion the refractive index of the glass =1.5). Thus the wave impulse from 
A reaches A ' at the same instant that the thrill from b reaches B. Similarly 
the impulses from A , b, and c reach the points A ", B', and C respectively, and 
the surface containing these points is the wave-front. When the impulse 
reaches H, it thrills A v ", B vl , C*, etc. at the same instant. The wave-front 
for the section represented in Fig. 10 is accordingly H, C v , B VI , A v " . . . H, 
which is no longer a simple circular arc. 
The ray directions or normals to the wave-front can then be readily found 
by drawing the secondary Huygens spherical wave-fronts as indicated in 
Fig. 10, and finding the points at which the wave-front is tangent to these 
circles. A simpler construction is that of Weierstrass as represented in Fig. 
ii, in which C is the center of the refracting sphere of radius R and refractive 
index n', while the refractive index of the enveloping medium is n. By con- 
