MR. R. A. SAMPSON: A NEM^ TREATMENT OF OPTICAL ABERRATIONS. 169 
Hence it is unnecessary to work out the expressions for focal lines and the rest 
afresh since they can all be Inferred without other change from what has already 
been given. 
Seidel’s five conditions are usually taken as the standard form for the conditions 
of existence of a correct normal image. We may follow these and express them in 
terms of the aberrational coefficients, proceeding pari passu with the two cases :— 
(1) Absence of spherical aberration 
(2) Absence of coma. 
(3) Absence of astigmatism . 
(4) Absence of distortion .... 
(5) A flat field, when (2) and (3) are 
satisfied. 
0' principal focus. 
= 0 
^yG = dj^H = 0 
= 0 
4H = 0 
O, O' conjugate foci, 
or = 0 
,, AH = ~ 0 
,, ~ 
,, Ad = 0 
AG = 0 
AH = 0 . (26) 
It is of interest to consider the position occupied by the well-known conditions 
usually quoted as “ Petzval’s condition for flatness of field,” and “ Abbe’s sine- 
condition.” 
Petzval’s condition, or = 0, we see from (18) to imply 
AG—ah = 0, Ad— ah = 0, ah—AI^ ~ hj AH—AL = 0, 
or, what is the same thing, simply 
Ad = ah and Ad = ah 
at all distances along O'x'. 
If we confine attention to the two cases above, we see that in the first, where the 
emergent origin is the principal focus, G == 0, and therefore Ad = AH without the 
intervention of = 0, and similarly in the second, when the emergent and original 
origins are conjugate normal foci, H = 0, and therefore Ad = AH ; the interpretation 
of these is the same, namely, that the comatic displacement is twice the comatic 
radius—comatic displacement” being used to denote the expression (AH-pAd) 
as on p. 168 —a well-known fact, usually put in the form that in the absence of 
astigmatism the successive comatic circles have two common tangents inclined to one 
another at 60 degrees. The other term remains as the true content of Petzval’s 
condition. Its interpretation may be put in diflerent forms ; as, apart from spherical 
aberration, at the normal focal plane of any image, the longitudinal axis of the focal 
ellipse is three times its transverse axis, which is an interpretation of the expressions 
of p. 166, for 8f = 0, 
Sb' = ... cos (fid [-|-cPAd4-|-;d"(Ad-l-2AH)], dc' = ... sin (pd [^d^Ad + A/3“Ad], 
of the first case, or the corresponding expression of the second case; or again, the 
distance of the focal circle beyond the normal focal plane is %f'ld x the radius of the 
VOL. ccxir.—A. 
z 
