On Systems of Rays. 103 
In the extensive case of a final uniform medium, the equation (B’) reduces itself 
to the following, 
oi ov pa ev ge a) ev 6B ‘ 
~ oa? dy cach \dy de 6p’ dx’ 
and, in the same case, the general conical locus of the second degree (#"'), connected 
with the condition of intersection of the final ray-lines, reduces itself to two real or 
(D") 
imaginary planes of vergency, represented by the quadratic 
= “ oy” + (=) oudy aE Sa’, (E*) 
and coinciding with the two planes of vergency considered in the sixteenth number : 
attending therefore to (C™), the relation (D”) may be geometrically enunciated by 
saying, that in a final uniform medium the two planes of vergency are conjugate 
planes of deflexure of any surface of a certain class determined by the nature of 
the medium, namely, that class for which, at the origin of co-ordinates, 
oz 28 Ow Oe COP: O82 BS 
ae tet = € 15 
Oa : da” dxoy . Sad” dy? a SB ” Le 
and therefore nearly, for points near to this origin, 
r ov Oy o2u 2 
z=per+yrts (2 +2ay sap tS im (G") 
ape 
the given final ray or axis of z being taken as the axis of deflexion, and the constants 
Ps 4, », being arbitrary. ‘This relation may be still farther simplified, by choosing the 
arbitrary constants as follows, 
; 1 ov af 1 &v aoe! * 
a ie a0 Soe (H") 
Z being any constant ordinate ; for then, (by the theory of the characteristic function 
V’ for a single uniform medium, which was given in the tenth number,) the surface 
(G°) acquires a simple optical property, and becomes, in the final uniform medium, 
the approximate locus of the points a, y, z, for which 
P, = fvds= Ve =const., ad") 
the integral 1” = /vds being taken here, in the positive direction, along the variable 
line , from the fixed point 0, 0, Z, to the variable point x, y, z, or from the latter to 
the former, according as Z is negative or positive. And though the equation (4G'”) 
is only an approximate representation of the medium-surface (J'’), which was called 
in the First Supplement a spheroid of constant action, and which is in the undulatory 
theory a curved wave propagated from or to a point in the final medium, yet since 
the equation (G’) gives a correct development of the ordinate z of this surface as far 
as terms of the second dimension inclusive, when the constants are determined by 
