108 Professor Hamitton’s Third Supplement 
direction of the path, and then to suppose this projection viewed directly from a final 
or initial distance determined as above. We might, for example, deduce from this 
theory the property of the guiding planes, the circular and elliptic appearances (B") 
(E") of the ellipse and circle (C'") (D"), and the acute or obtuse appearance (X'"’) 
of a right angle in the final perpendicular plane, when the directions of the sides of 
this angle are different from those of the object-axes of distortion. And the relations 
(M”) for extreme ordinary media may be expressed by the following theorems: 
first, that the angle (2G) between the final pair of planes of no distortion (Y"’), 
is equal to that between the initial pair (Z"); second, the visible angular mag- 
nitudes of any small and equal linear objects in final and initial planes of no dis- 
tortion, are proportional to the indices of the final and initial media, when the 
objects are viewed along a given luminous path, from the initial and final points ; and 
third, the two intersection-lines of the two pairs of planes of no distortion coincide 
each with the visible direction of the other, when viewed along the path. 
Calculation of the Elements of Arrangement, for Arbitrary Axes of Co-ordinates. 
22. In the foregoing formule for the elements of arrangement of near rays, we 
have chosen for simplicity the final and initial points of a given luminous path, as the 
respective origins of two sets of rectangular co-ordinates, final and initial, and we 
have made the final and initial ray-lines, or tangents to the given path, the axes of 
z and 2’; a choice of co-ordinates which had the convenience of reducing to zero 
eighteen of the forty-two general coefficients in the expressions of da, 8B, dy, da’, 88,87, 
as linear functions of 8x, dy, dz, da’, dy’, 62’, 8x. The twenty-four remaining coeffi- 
cients (D*) may however be easily deduced, by the methods already established, and 
by the partial differential coefficients of the characteristic and related funtions, from 
other systems of final and initial co-ordinates, for example, from any other rectangular 
sets of final and initial axes. 
In effecting this deduction, it will be useful to distinguish by lower accents the par- 
ticular co-ordinates and cosines of direction, which enter into the expressions (D*), 
and are referred to particular axes of the kind already described ; and then we may 
connect these particular co-ordinates and cosines with the more general analogous 
quantities « y za’ yz aB ya fy, by the formule of transformation given in the 
thirteenth number, which may easily be shown to extend to the case of two distinct 
rectangular sets of given or unaccented co-ordinates. In this manner the axes of 
z, and 2’, considered in the thirteenth number, become the final and initial ray-lines, 
and we have, by (4°), 
