On Systems of Rays. 101 
Relations between the Elements of Arrangement, depending only on the Extreme 
Points, Directions, and Colour, of a Given Luminous Path, and on the Extreme 
Media. Ina Final Uniform Medium, Ordinary or Extraordinary, the two Planes 
of Vergency are Conjugate Planes of Deflexure of any Surface of a certain class, 
determined by the Final Medium ; and also of a certain Analogous Surface, 
determined by the whole combination. Relations between the Visible Magnitudes 
and Distortions of any two small objects, viewed from each other through any 
Optical Combination. Interchangeable Eye-axes and Olyect-axves of Distortion. 
Planes of No Distortion. 
21. It was shown in the fourteenth number, and the result has since been developed 
in detail, that the general geometrical relations between the extreme directions of 
infinitely near rays are determined by the co-efficients of the linear variations éa, 83, dy, 
da’, 68’, dy’, of the six marks of extreme direction, considered as functions of the six 
extreme co-ordinates and of the colour; and that, between the forty-two general 
coefficients of these six linear variations, there exist eighteen general relations, leaving 
only twenty-four coefficients arbitrary, if we suppose for simplicity that the final and 
initial co-ordinates are referred to rectangular axes. But besides these eighteen 
general relations which are common to all optical combinations, there arise certain 
other relations between the coefficients, when the extreme media are considered as 
given, and when the extreme points, directions, and colour, of any one Juminous path, 
are also supposed to be known. For, if we then employ the general equations (4°), 
we may consider the extreme medium functions v, v’, and their partial differentials, 
as known, and may deduce general expressions for the coefficients before mentioned 
of the linear variations of the extreme cosines of direction, involving only, as 
unknown quantities, twenty-seven partial differentials of the second order of the cha- 
racteristic function 7, namely, all of this order, which are not relative to the variation 
of colour only ; but these twenty-seven are connected by the fourteen general rela- 
tions (Q) (U) (X) (Y), deduced in the third number, of which however only thir- 
teen are distinct, because the two systems (U) (Y) conduct both to one common 
equation (D) ; there remain, therefore, as independent quantities, only fourteen of 
the partial differentials of V7, in the general expressions of those twenty-four coeffi- 
cients of the linear variations of the extreme direction-cosines, which had before 
been considered as independent, when the extreme medium-functions v, v’ were sup- 
posed unknown and arbitrary : and if we eliminate the fourteen independent differen- 
tials of V” between the expressions of these twenty-four coefficients, we shall obtain 
ten general relations, between the elements of arrangement of infinitely near rays, 
VOL. XVII. 2D 
