162 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
for two surfaces Bq, Bg with refractive indices, /x_i, /X 3 , as in Seidel’s convenient 
notation 
+ = M-l 
[(---) 
L>i /u_i/ 
V3 Ml/ j 
and for any sequence of surfaces whatever 
— ^)b,,. . . 
'MSr + l 
This will be recognized as the expression which figures in the well-known “ Petzval 
condition for flatness of field.” It was given by Petzval without proof in 1843, and 
it is a comment upon the difliculty which the geometricaHnethod finds in removing a 
condition that may have been tacitly introduced that its proper position has so far 
remained obscure. Its general geometrical implications will be considered later. 
Besides the condition that the rays of any thin bundle should always be normal to 
a surface there is another general property to which they are subject in all systems. 
For normal systems in which we have stigmatic correspondence this is usually called 
the Helmholtz magnification theorem connecting the linear and angular magnifi¬ 
cations. For aberrational systems it would at first appear as if both linear and 
angular magnifications lost their meaning, but I have succeeded in generalizing the 
theorem in the paper already referred to.* In the first place focal lines in the 
original system are shown to correspond one to one and not pair to pair with focal 
lines in the emergent system ; and I'ays which issue from any point in a focal line in 
a plane perpendicular to that line lie in a plane in the emergent system perpendicular 
to the conjugate focal line which they meet in a point. Such planes are called planes 
of correspondence. The behaviour of any ray may be traced through the behaviour 
of its projections upon the planes of correspondence. The separation of two parallel 
focal lines compared with the separation of their two conjugates preserves the idea of 
linear magnification and the angles in the planes of correspondence that of angular 
magnification. Then if a is the separation of two focal lines which lie parallel to one 
another in a plane perpendicular to an original ray at any point and a! that of their 
two conjugates, and if a is the angle between two rays issuing from one of these lines 
in a plane of correspondence perpendicular to both and a! the angle between the 
same rays on emergence, it is proved that 
fxOba. = fxCi' a!. 
This is completely general. Now return to the case of surfaces centred upon an 
axis. It is clear that for any point olF the axis, say the point (O, h, O), one of the 
planes of correspondence, is the meridianal plane passing through the axis and the 
point itself. 
* ‘ Proo. E. M. S.,’ vol. 29, p. 70. 
