104 Professor Hamitton’s Third Supplement 
(H"), the conclusion respecting the deflexures applies rigorously to the surface (7°); 
and the two planes of vergency (HE), in a final uniform medium, are conjugate 
planes of deflexure of the spheroid or wave (I). We shall soon resume this result, 
and endeavour to illustrate and extend it. In the mean time we may remark that the 
same planes of vergency (H") are also conjugate planes of deflexure of a certain 
analogous surface, determined by the whole combination, and not merely by the final 
uniform medium, namely, the surface (D"), for which 
JSvds (= V )= const., (K") 
the integral being here extended to the whole luminous path, and being therefore 
equal to the characteristic function V” of the whole optical combination ; an additional 
property of the planes of vergency, which is proved by the following relation, analo- 
gous to (D"), and deducible from (C*) or (G@*), 
BF 3 EY (38 _Ba) _ BY 8 
= 3a oy | Sxdy Sy Sy de' 
Sy be ce 
Finally, with respect to the four remaining equations, of the third groupe (C), it 
is evident that they express certain general relations depending on the extreme media, 
between the coefficients which determine the guiding planes and conjugate guiding 
axes, for the final and initial ray-lines. In the extensive case of extreme ordinary 
media, they reduce themselves to the four following, which may also be deduced 
from (F'*), 
(M") 
nu, « being the indices of the media; and they conduct to some simple conclusions, 
respecting the general relations between the visible magnitudes and distortions of a 
small plane object, placed alternately at each end of any given luminous path, and 
viewed from the other end, through any ordinary or extraordinary combination : at 
least so far as we suppose these distortions and magnitudes to be measured by the 
shape and size of the initial and final ray-cones. For then the conjugate guiding 
axes, initial and final, perpendicular to the given path at its extremities, and deter- 
mined in the fifteenth number, may be called the eye-axes and olject-axes of distor- 
tion, for a small object placed in the final perpendicular plane, and viewed from the 
initial point ; and if we take these for the axes of initial and final co-ordinates, so 
as to have, by (X") (Y™), 
ba’ 6B" 83" da’ 68’ 
= 0, =O, > 0, eg s 
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