106 Professor Hamiiton’s Third Supplement 
of the extreme ordinary media, the two perpendicular sections thus obtained will be 
equal and similar to each other ; and if, besides, we put, by ( Y"), 
See = cos. G, (S*) \ 
( & being by (#"") the inclination of an initial guiding plane to the plane perpendicular 
to the given initial ray-line,) and determine also the arbitrary quantity # as follows, 
wl 8a’ —1_—-!1 oa —] 
a aay Ge) = i lecaliovi2 (T”) 
the perpendicular sections of the initial and final ray-cones may then be represented 
as follows, 
, : , ci : 
U2, ChyiG@y = = G) - (U0) 
and 
Y éa\ —1 15 
y=Ccos, Gafi();) 2 = (3) : CV?) 
the visible distortions therefore, depending on the inclination G, are the same for any 
two small equal objects, thus perpendicularly and similarly placed at the ends of any 
given luminous path, and viewed from each other along that path, through any 
optical combination. 
The distortion here considered will in general change, if the object at either end of 
the given luminous path be made to revolve in the perpendicular plane at that end, so 
as to change its position with respect to the axes of distortion, For example, if the 
object be a small right-angled triangle in the final perpendicular plane, having the 
summit of the right angle at the given final pomt B of the path, we know, by the 
theory given in the fifteenth number, that the right angle will appear right to an eye 
placed at the initial point 4, when the rectangular directions of its sides $4, $25 
coincide with those of the final guiding axes, or object axes of distortion ; but that 
otherwise the right angle ¢’,—¢', will appear acute or obtuse, its apparent magnitude 
¢2— 4 being determined by the formula 
T sl 7 . 
= tan. | ¢:—¢ —-) = —_—__ Sin: 264, wae) . 
CaaS ae 
ba Oy 
which may, by (S"), be reduced to the following, 
— tan. ( po— bi =) = dsin. G. tan. G. sin. 2 ¢). (X*) 
The law of change of the distortion, corresponding to a rotation in the final perpen- 
dicular plane, may also be deduced from the theory of the guiding planes, explained 
in the fifteenth number. 
“ww 
