170 MR. R. A. SAMPSON; A NEW TREATMENT OF OPTICAL ABERRATIONS. 
focal circle. This sliows the connection with flatness of field, hut to restrict the 
reference to curvature of the field is a misconception of the significance of Petzval’s 
condition. If Petzyal’s condition holds, the result stated above is true for all normal 
image planes. 
Abbe’s sine-condition for absence of coma states that if the magnification produced 
by ra 3 ^s passing between two conjugate foci through all zones is the same, the relation 
must hold 
sin O'/sin 0 = const.. 
where S, 0' are the original and emergent inclinations of the ray to the axis, 
notation this would run 
or, if 
it gives 
= const .; 
/3' = (L + ^L)/3 
^L/L = i(L^-l)/3=^; 
In our 
now we have seen on p. 163 that the linear magnification is 
(N + M)/(L + <5L) 
and the condition this should be constant is 
SL/L = <5N/N, 
and this, in accordance with p. 163, gives the conditions 
S,L/L:=K\ S,L/L = KL, ^L/L = L^-1. 
To make these agree with the sine-condition we must take 6 = 0, so as to remove 
(^iL, AL from the reckoning. We see then that such an assumption underlies the 
application of the sine-condition. 
I shall next show how these formulae may be applied to the numerical calculation 
of lenses. For this they are particularly appropriate if the calculations are made 
with any ordinary type of multiplying machine and not with logarithms. To the 
best of my judgment they appear to require a fraction only of the work involved in 
the equivalent complete trigonometrical calculation and, as will be shown, they are 
certainly not less accurate for telescopic object glasses. They show with remarkable 
clearness the contribution of each surface to each fault of the image. They also 
supply throughout their course a number of natural checks upon the computation 
which are searching and usually complete. 
I shall take as my example the celebrated object glass of the Fraunhofer heliometer 
at Konigsberg. This is a small lens of aperture 6'2 inches and focal length 101 inches 
which was constructed by Fraunhofee. Bessel, in describing the heliometer and 
its corrections, vuth his customary * masterlj" thoroughness, calculated this lens 
