the Five Aberrations of von Seidel. 685 
condition by elementary methods will be found in Whittaker's 
tract *. 
Before leaving systems symmetrical about an axis to which 
all the rays are inclined at small angles, we may remark 
that, as U (4) contains 6 constants, in like manner U (6) 
contains 10 constants f and U (8) 15 constants, of which in 
each case one is ineffective. 
The angle embraced by some modern photographic lenses 
is so extensive that a theory which treats the inclinations as 
small can be but a rough guide. It remains true, of course, 
that an absolutely flat field requires the fulfilment of the 
Coddington-Petzval condition ; but in practice some com- 
promise has to be allowed, and this involves a sacrifice of 
complete flatness at the centre of the image. It will be best 
to fulfil the conditions da:/dl = 0, dy/dm = 0, or, what are 
equivalent, 
d?U/dl 2 = 0, d 2 U/dm 2 = 0, 
not when I is very small but when it attains some finite 
specified value. If we suppose ,y' = 0, U is a function of 
x' 2 , l 2 + m 2 , and lx' : or say of u, v, w. Hence 
dl 2 dv 2 dv die 2 dw dv ' 
dm 2 dv 2 dv ' 
After the differentiations are performed, we are to make 
m = ; so that the two conditions of astigmatism and focus 
upon the plane, analogous to (28), (29), are 
f = o, tf fS + ^ *u 
dv dv 2 dw 2 dv dw 
in which v is to be made equal to I 2 . But it is doubtful 
whether such equations could be of service. 
Let us now suppose that the system is indeed symmetrical 
with respect to the two perpendicular planes of x and y, but 
not necessarily so round the axis of z. In the expression 
for U no terms can occur which would be altered by a 
simultaneous reversal of x' and I, or of y' and m. For U (2) 
we have 
U< 2 > = al 2 + £m 2 + ytfl + Sy'm 
+ terms independent of I and m. 
* 'Theory of Optical Instruments/ Cambridge, 1907. The optical 
invariants, introduced by Abbe, are there employed. 
f Schwarzschild, loc. cit. 
