On Systems of Rays. 9 
as depending, by (2), for all optical combinations, on the seven quantities a7 vy z x. 
In like manner, we shall consider the new auxiliary function JZ’ as depending, by the 
new transformation (C’), on the seven quantities rv o'r vy. The forms of these 
auxiliary functions, V7, 7, are connected with each other, and with the characteristic 
function V7, by relations of which the knowledge is important, in the theory of 
optical systems. Let us therefore consider how the form of each of the three func- 
tions, V, WW, T, can be deduced from the form of either of the other two. 
These deductions may all be effected by suitable applications of the three forms 
(41) (B) (C), of our fundamental equation (4), together with the definitions (D’) 
(E), as we shall soon see more in detail, by means of the following remarks. 
When the form of the characteristic function /” is known, and it is required to 
deduce the form of the auxiliary function //’, we are to eliminate the three final 
co-ordinates, x, y, z, between the equation (J) and the three first of the equations 
(E£); and similarly when it is required to deduce the form of ZT from that of V, 
we are to eliminate the six final and initial co-ordinates x y z x yz between the six 
equations (#), (which are all included in the formula (4’),) and the following, 
T=—-V+a0-270 tyr —yr +zu—Zu : (F) 
and if it be required to deduce the form of ZY from that of WV, we are to eliminate 
the three initial co-ordinates 2’ yz’, between the equation (’) and the three follow- 
ing general equations, 
7 
Se ae : (G’) 
But when it is required to deduce reciprocally V from T or from /V, or W from T, 
we must distinguish between the cases of variable and of uniform media; because we 
must then use the equations into which (B’) and (C’) resolve themselves, and this 
resolution, when the extreme media are not both variable, requires the consideration 
of the connexion that then exists between the quantities oz vo' 7 vy: which circum- 
stance also, of a connexion between these variable quantities, leaves a partial indeter- 
minateness in the forms of Zand JV as deduced from V7, and in the form of J’ as 
deduced from WV, for the case of uniform media. 
When the final medium is variable, then o,z,v, x, may in general vary indepen- 
dently, and the equation (B’) gives 
oo (H’) 
and, in this case, Y can in general be deduced from WV by eliminating o, 7, v, between 
the equation (D’), and the three first equations (H’). But if the final medium be uni- 
form, then o, 7, v, xX» are connected by the first of the relations (#’), from which, in this 
case, the final co-ordinates disappear ; and instead of the four equations (H’) we have 
the three following 
VOL. XVII. D 
