V. A New Treatment of Optical Aberrations. 
By li. A. Sampson, F.B.S. 
Received March 15,—Read May 23, 1912. 
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The method developed by Gauss in liis ‘ Dioptrische Untersiichimgen ’ is probably 
the most powerful, as well as the readiest, method in geometrical optics. It has 
in effect hitherto been restricted to S 5 ^stems in which the relations of oiiginal and 
emergent rays are strictly linear, or, in optical language, those in which the 
aberrations can be neglected. It is true that Seidel bases his celebrated discussion 
of aberrations upon Gauss’s method, but he soon modifies it and replaces its system 
of co-ordinates and characteristic steps by others. The following pages show how tlie 
method may be extended and retained throughout the discussion of the aberrations of 
any co-axial system. They will be found to throw light upon the general relation¬ 
ships of the well-known Petzval condition and Abbe Sine condition, to furnish a ready 
method of describing, analysing and measuring the faults of an optical image, and to 
be particularly adapted to numerical calculations, to the order to which tliese are 
necessary for telescopic objectives. 
It will be convenient to state here the essentials of the method in the form in 
which they will be used later. Let Oxyz, O'x'y'z' be rectangular axes in the original 
and emergent media, of which the refractive indices are fx, T respectively. f).r, O'x 
are the axes of the optical system. Take the equations of any ray l)efore and after 
its passage through the system in the respective forms 
y = I3x + t>, z — yx + c, 
and .( 1 ) 
y' = fx' + ty, z' = fx' + c', 
then, provided there is a strict linear correspondence as well as symmetry about the 
axis, we may put 
}/ — p — gfc + Ay, 
.( 2 ) 
f — kh + l^, y — /.'C-e/y, 
VOL. QCXII.-A 488. Publialied separutely, July 27, 19] 2. 
