[ 29 ] 
II. On a Theory of the Second Order Lonyitudinal Spherical Aberration for 
a Symmetrical Optical System. 
By T. Y. Baker, B.A., Instructor Commander, R.N., and L. N. G. Filon, M.A., 
D.Sc., F.R.S., Goldsmid Professor of Applied Mathematics and Mechanics in 
the University of London. 
Received December 2, 1919,—Read February 12, 1920. 
§ 1. Statement of the Problem and Historical References. 
If we consider a pencil of rays issuing from a point on the axis of a symmetrical 
optical system (i.e., a system of refracting spherical surfaces, the centres of which lie 
on a straight line called the axis of the system), it is well known that, if the pencil 
be a thin one, of which the mean ray is along the axis, the first approximation to the 
emergent pencil is another punctual pencil, of which the rays pass through an image 
point, also situated on the axis. The general method of treatment of such image 
points, which are usually referred to as “geometrical” images, is due to Gauss, and is 
developed in any text book of Geometrical Optics. 
When, however, the pencil considered is one of finite aperture, the outlying rays 
do not, after emergence, pass through the Gaussian image point, nor do they have the 
inclination assigned to them by the Gaussian calculation. The emergent rays lying 
in any one axial plane touch an envelope or caustic, which has one cusp at the Gaussian 
image, with the axis as proper tangent. The intercepts of any given emergent ray 
upon the axis and the image plane, measured from the Gaussian image, are known as 
the longitudinal and transverse spherical aberrations of that ray. 
It is clear that if both these spherical aberrations, or either of them together with 
the inclination of the ray on emergence, be known for every possible position of object 
and image, and for every possible inclination of the incident ray, the whole complex 
of emergent rays lying in axial planes can he mapped out. The calculation of these 
aberrations is therefore of fundamental importance in practical optical design, where 
we do not deal with infinitely thin pencils. 
The method employed hitherto for dealing with aberrations from the mathematical 
standpoint has been to develop the sines occurring in the refraction equations at each 
spherical surface in ascending power of some argument, which may be either the 
circular measure, or the sine, or the tangent, of one of the angles concerned, and to 
calculate, by the usual methods of successive approximation, the required aberrations 
as a series of ascending powers of such argument. 
VOL. CCXXi. A 583. F [Published June 16, 1920. 
