30 
MR. T. Y. BAKER AND PROF. L. N. G. FILON: LONGITUDINAL 
When this is done it is found that the terms due to the first power of the argument 
lead to the Gaussian image point, so that the series begin with a term involving the 
second power of the argument in the case of the longitudinal aberration, and the 
third power in the case of the transverse aberration. These terms are the first order. 
The following terms next in sequence, which are of fourth and fifth power respectively, 
are spoken of as aberrations of the second order, and so on. 
A considerable amount of theoretical work has been done on aberrations of the 
first order by Seidel, Abbe and others, and the treatment of these is fairly well 
known. Unfortunately, it is found in practice that the first order aberrations do not 
give a sufficient approximation for the optician’s requirements. In fact for a certain 
range of object and image positions, they are so badly out that they cannot be 
said to constitute an approximation at all. This fact has long been recognised by 
optical designers, whose practice is invariably to calculate, using the exact trigono¬ 
metrical equations which involve no approximation at all, the correct paths of a 
number of selected rays, from which they draw conclusions as to the efficiency, or 
otherwise, of the proposed system from the practical point of view. 
The trigonometrical method, however, from the designer’s point of view, has the 
radical defect that, while it gives partial information about the performance of a given 
system, it gives no direct intimation of the direction in which the elimination of various 
defects is to be looked for, and it entails a long and laborious process of seeking for 
the optimum by trial and error. 
The object of the authors of the present paper has been to develop a method of 
expressing the aberrations, which, while carrying the algebraic development to a 
stage including the second order, should be free from certain grave troubles involved 
by failure of convergency, troubles which appear to have been hitherto neglected. 
In fact this method gives numerical results that, for a single lens, are considerably 
more accurate than the ordinary second order formulae. Further, these methods 
enable one to deal, in a comparatively easier form, with the problem of the second order 
aberrations of combinations of surfaces and systems, a problem which, so far as we 
know, has never been attacked from any general standpoint. Koenig and Yon Rohr 
(Yon Rohr, ‘ Theorie der Optischen Instrumente,’ Cap. Y.) give a development of a 
formula for the coefficients of first order and second order in the longitudinal spherical 
aberration, based on Abbe’s method of Invariants, but so far as can he seen, no definite 
results are obtained for the second order terms. 
Dennis Taylor (‘System of Applied Optics,’ p. 67) gives a formula for the 
spherical aberration, developed in powers of the intercept made by the ray on the 
first principal plane, which includes terms of second order. But his formula, a 
particular case of those dealt with in the present paper, is limited to the thin lens, 
and no attempt seems to be made at anything like a general treatment of such 
aberrations. 
Another important object of the method to be described is to express the 
