12 Professor Hamitton’s Third Supplement 
V, W, T, may be summed up in this one rule or theorem: that each of these three 
functions may be deduced from either of the other two, by using one of the three 
equations (D’) (Z’) (F") and by making the sought function a maximum or minimum 
with respect to the variables that are to be eliminated. For example we may deduce 
T from V’, by making the expression (#") a maximum or minimum with respect to 
the initial and final co-ordinates. 
An optical combination is more perfectly characterised by the original function /’, 
than by either of the two connected and auxiliary functions 7, 7’; because 
enables us to determine the properties of the extreme media, which /V and J’ do not; 
but there is an advantage in using these latter functions when the extreme media are 
uniform and known, because the known relations which in this case exist, of the forms 
(K’) and (N’), (together with the other relations (@Q") which arise when the combi- 
nation is prismatic,) leave fewer independent variables in the auxiliary than in the 
original function. At the same time, as has been already remarked, and will be after- 
wards more fully shown, the existence of relations between the variables produces a 
partial indeterminateness in the forms of the auxiliary functions, from which the 
characteristic function 7 is free, but which is rather advantageous than the contrary, 
because it permits us to introduce suppositions and transformations, that contribute 
to elegance or simplicity. 
General Transformations, by the Auxiliary Functions IV, T, of the Partial Dif- 
ferential Equations in V. Other Partial Differential Equations in V, for 
Extreme Uniform Media. Integration of these Equations, by the Functions W, T. 
5. Another advantage of the auxiliary functions W, 7, is that they serve to 
transform, and in the case of extreme uniform media to integrate, the partial differ- 
ential equations (C), which the characteristic function /” must satisfy. In fact, if the 
final medium be variable, the first of the two partial differential equations (C) may 
be put by the foregoing number under the two following forms, 
SW 8W 8W 
OSD (erry es Ee) 
da * or” ou T 
0=2( ee: ce) 
= Oy Ts Vy Sau?) eae 32% 5) 
and if the initial medium be variable, the second of the two partial differential equa- 
tions (C’) may be put under these two forms, 
ao QW WW 
o=o'(—, ay” 3” a,y',2,X); U’ 
Sg aloe oy) W 
> 
So Se a 
Obsa; (¢, r, Vv, ai 
