166 ME. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
of wliich the former may be called the comatic increase of focal length and the latter 
the distortional increase of focal length , since the.se are evidently their characters. 
The two terms 
cos <pd[K,f' + + {§,G + 2S.fl)l sin ^i>d[K^f' + ^d%G + ^^%G'] 
represent an ellipse which may be varied by choosing cf' at different values. If we 
take 
the ellipse becomes the primary focal line ; if we take 
it becomes the secondary focal line, in advance of the primary line by the amount 
Generally I shall call it the foccd ellipse and, as a rule, shall take 
Klf'+U%G + ^/3‘^{S.jG + S,E.) = 0, 
which gives the focal circle 
d cos (j) —d sin (p 
situated midway between the focal lines. This circle is described backwards as the 
original circle d — const, is descriljed forwards. 
Finally the terms 
(.V cos 2^ [l-zdc^aG], d"^ sin 2<p 
give another circle which I shall call the comatic circle; its radius = x co/?iatm 
increase of focal length, so that they vanish together. As the original circle 
d = const, is described once, forward, it is described twice, forward, each point upon 
it corresponding to two diametrically opposite points of the original circle. 
Consider the focal circle and the comatic circle simultaneous!}^; we may take 
I cos <p-{-ni cos 20, —I sin 0 + 7>? sin 20, 
where 
I = m = g-cZ^/S^G; 
this is a trochoidal curve, which l)ecomes a three cusped hypocycloid for I = 2m and 
goes through the types illustrated below for different values of /3/d. For a given 
value of /3 all these types are present for different values of d, and are described 
about different centres owing to the comatic increase of focal length. These facts are 
well known in particular instances, and even experimentally, but as far as I can find 
they have not hitherto been expressly demonstrated generally. 
The plane at which these phenomena are found is taken at x' — Sf', where 
Sf'/f' = -KSf' = id%G + ^d^{S,G-^SM). 
