On Systems of Rays. 107 
The distortion will also change, if the small plane object be removed into an oblique 
instead of a perpendicular plane. In this case we may still employ the equations 
(A") (0) for the initial and final ray-lines, and may still represent the initial and 
final ray-cones by the equations (47") (Q") ; but we are now to consider the equa- 
tions (@") (P"), for the final and initial objects, as representing the projections of 
those objects on the extreme perpendicular planes ; or rather the projecting cylinders, 
which contain the objects, and which determine their visible magnitudes and distor- 
tions, by determining the connected ray-cones. For example, the equation (C"') may 
be considered as representing a final elliptic cylinder, of which any section near the 
final point B of the given luminous path will correspond to an initial circular ray- 
cone (B"), and will therefore appear a circle to an eye placed at the initial point 4 ; 
while on the other hand we may regard the equation (D") as respecting a final 
circular cylinder, such that any section of this cylinder, near the final point B, will 
give an initial elliptic ray-cone ("'), and will appear an ellipse at 4. And as 
the elliptic ray-cone (#") conducted, by its circular sections, to the guiding planes 
(F") for the initial ray-lines, so, for small plane final objects, the planes 
z=+z2 tan. G, CY?) 
namely, by (S"), the planes of circular section of the elliptic cylinder (C™), are 
planes of no distortion ; in such a manner that not only, by what has been said, the 
circular sections themselves in these two planes appear each circular, but every other 
small final object in either of the same two planes appears with its proper shape to an 
_eye placed at the initial point A of the given luminous path ; the angular magnitude 
of the final object thus placed, being the same as if it were viewed perpendicularly by 
straight rays, bik ae refracting or reflecting surface or medium interposed, from 
ay ) —1_ In like manner, the planes 
a final distance = ( 
Zee. tan. Cr (Z') 
which are the planes of circular section of an analogous initial elliptic cylinder, are 
initial planes of no distortion, of the same kind as the final planes (Y'); since any 
small initial object, placed in either of these two initial planes (Z"), and viewed from 
the final point B of the given luminous path, will appear with its proper shape, and 
with the same angular magnitude as if it were viewed directly from an initial distance 
i] SBE nl efoo y= 1 
es SUE ae 
This theory of the planes of no distortion gives a simple determination of the 
visible shape and size of any small object placed in any manner near either end of a 
given luminous path ; since we have only to project the object on one of the two 
planes of no distortion at that end, by lines parallel to the corresponding extreme 
