168 MR. R. A. SAMPSON: A NEW TREATMENT OF OPTICAL ABERRATIONS. 
d = //SO. 
d = 
mo. 
d = 
mo. 
h’. 
15'. 
30'. 45'. 
60'. 
15'. 
30'. 
45'. 
60'. 
15'. 
30'. 
45'. 60'. 
Focal radius . 
0"-06 
0"-26 0"-59 
l"-05 
0"-10 
0"-40 
0"-89 
l"-57 
0"-19 
0"-79 
l"-77 ! 3"-14 
Comatic radius 
0"-25 
0"-50 0"-75 
l"-00 
0"-56 
1"-13 
l"-69 
2"-25 
2"-25 
4"-50 
6"-75 9"-00 
Comatic mag- 
nification . 
1■0005 
1-0005 1-0005 
1-00051-0012 
1-0012 
1-00121-0012 
1-00501-0050 
1-00501-0050 
' Comatic dis- 
placement . 
- 
0"-50 
l"-00 l"-50 
2"-00 
1"-13 
2"-25 
3"-38 
4"-50 
4"-50 
9"-00 
13"-50,18"-00 
The last two lines measure the same thing, the fourth representing 
in angle, and the third 1+|-(JiH + r^aG) d^/3//^', and owing to the relation 4^1 = 4H 
the second line contains quantities one-half that of the fourth. Since these 
aberrations stand uncompensated, it is clear that the statement often made that the 
reflector has a very limited field is fully borne out, especially when, as is often the 
case, the ratio of semi-aperture to focal length is so great as l/lO. In this case, at 
only 30' from the centre of the field, the light which comes from the outermost zone 
of the mirror would be spread around a little ring which was nearly a circle of 10" 
diameter, having its centre 9" from the correct normal position for the image. 
Turn now to the other case which was proposed for discussion on p. 165, namely, 
where the original 0, O' are conjugate foci, so that H = 0 and 4Gr = 4H. We have 
to study the delineation of any point in Oyz upon the plane O'y'z! or planes close 
to it. We may take the point h — const., c = 0, and then make /3, y vary, so that, 
e.g., for /3^-t-y“ = const, the ray through the point {h, O) describes a cone with axis 
parallel to Oa;. Let us put 
/3 = cos \}/-, y = 0 sin xf/-, 
and we have, at the plane x' = Sf'. 
h' + 8V = + + +W']/3 
+ 4- cos \/^4Gr 4- 0‘4O] 4- ^6 cos xp- + 2h6 cos 4- 
c'-h.5c'= [ =" +W']r 
4- * 4-^fisin^/^[6^4H4-26ficosi/r4H4-0“4H].(25) 
Compare these with the expressions (23) of p. 165, and we see that they run upon 
exactly the same model, but with a change of o'dle in which we replace 
by 
4G, 
4G, 
4G, 
4H, 
4H, . 
d, (p, 
4H, 
4H, 
4H, 
4G, 
4G, 
4G, 
0, xj^, 
h. 
