SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
71 
We have further obtained a method for calculating the coefficients of this formula 
for any symmetrical optical system in terms of the coefficients for the components, 
in such a way that the effect of any single component upon the whole combination is 
immediately obtained. 
In considering the convergency of the series usually employed, we have found that 
the value of the approximation depends upon the particular variable employed, and 
that if we wish to avoid trouble owing to lack of convergency we must use sin a u or 
tan a 0 (or a suitable multiple of these) as argument, where a 0 is the inclination of the 
original incident ray. 
The numerical success of the new formula appears to suggest that progress in the 
algebraic treatment of symmetrical instruments is to be sought, not so much along 
the lines of developments in series, but in other mathematical directions such as 
continued products, or possibly continued fractions. 
The next step would be to develop the method so as to cover the second order 
approximations to the emergent inclination. This will enable us to deal with aberra¬ 
tions off the axis of the system. 
Some progress has already been made by the authors in this direction and the 
results will, it is hoped, form the subject of a later communication. 
PRESENTED 
23,m (320 
