36 
MR. T. Y. BAKER AND PROF. L. N. G. FILON : LONGITUDINAL 
the maximum a 2 tend to zero, so that only an infinitesimal portion of the rays can be 
dealt with by the method of successive aberrations, i. e. , by the Taylor’s series. 
That the range of failure is by no means an unimportant one is shown by an 
example given by the authors in a paper read before the Optical Society in December, 
1918. In this example the system considered is a positive lens of unit focal length 
and thickness y'g, meniscus shaped, with curvatures 1 and 2'36, and its convex side 
towards the incoming light. For such a lens and magnification as low as 2, the 
critical value of a i is found to be about 4° 40', corresponding to a value of a 0 of 13°, 
whilst the greatest practical value of a 0 is 26°, so that in this case only about \ of the 
light going through the lens could be dealt with by series in terms of the emergent 
angle. From M = 2 to M = oo the conditions are still worse. 
As a matter of fact, it appears that in this case the range of magnifications, within 
which development in terms of the emergent inclinations is possible for all rays 
travelling through the lens, is restricted to a range lying somewhere between M = — 1 
and M = 1'5. This makes it clear that we cannot depend, in the calculation of the 
aberrations of an optical system, upon any series with the emergent inclination as 
argument. This is important, because from other considerations it would have been 
valuable to have been able to express the equation of the emergent ray in the 
form 
y + qx =/{q) 
where q is the inclination of the emergent ray, and to proceed to obtain successive 
approximations to the caustic by developing f (q) in powers. It now appears that 
this is not, in general, legitimate. 
We now come to the consideration of series proceeding by powers of a 0 , or of its 
trigonometrical functions. Here the question of many-valuedness will not occur, 
except as follows. 
If we consider a ray impinging upon a spherical refracting surface, this ray, if 
produced, will meet the surface at a second point. Treating the problem from the 
purely analytical standpoint, this second point is also one at which refraction takes 
place, and thus, for the same a 0 , there will, in general, be two values of a 2 , four of a 4 , 
and so on. a 2n will therefore, in general, be a multiple-valued function of a 0 , and 
the aberrations will also be multiple-valued functions, and the branch-points of these 
multiple-valued functions will, as before, limit the convergency of the Taylor 
series. 
Now clearly two branches coincide whenever there occurs a grazing incidence ; 
and, therefore, if the system be so arranged (as it almost necessarily is) so that no 
grazing incidence is reached, there will be no real branch-points within the range of 
practical values. But this does not mean that the Taylor’s series will necessarily 
be valid, for there might be imaginary branch-points. A very simple example will 
show how such branch-points can occur. 
