6 Professor Hamitton’s Third Supplement 
(which are of the second order, because a, es y, @, By, are defined by the equations 
vida =- 
= ds P ae a ee 
dz’ ijn te (P) 
= ik jy? a= a “e i 
a= 
the symbol d referring, throughout the present . to motion along a ray, 
while 8 refers to arbitrary infinitesimal changes of position, direction, and colour, and 
ds’ being the initial element of the ray,) were deduced, in the First Supplement, by the 
Calculus of Variations, from the law of least action. The same forms (0), which are 
equivalent to but two distinct equations, may be deduced from the fundamental 
formula (4), by the properties of the characteristic function /. For, if we differ- 
entiate the first equation (C), (which involves the coefficients of this function V, 
and was deduced from the formula (.4),) with reference to each of the three co-ordi- 
nates, x, y, z, considered as three independent variables, and with reference to the 
index of colour y, we find, by the foregoing number, 
OV Or eV _ dv 
aye + Pacey t 1 Bebe Se? 
eV ae ct eV kv 
* Scat P aye + 7 Syde— ay’ 
lla Pitas SONG 2 att 2 (Q) 
se yes PY Bsa 
eV Cada eV ov 
” 8x8 ae Bydy | Yd dy? 
and the three first of these equations (@Q), by the help of the general relations (B), 
which were themselves deduced from (4), and by the meanings (P) of a, B, y, may 
easily be transformed to (0). The differential equations (QO) may also be regarded 
as the limits of the ere 
=) e-fe@)u-vs@), = ® 
oy 
= saad 
are obtained by differentiating ~ considered as a function of the seven variables 
®, Y, % Ax, Ay, Az, x, if Av=ax—a', Ay=y—y', Az=z—Z; the variation of V, 
when so considered, being by (4), and by the definitions (#7), 
aT &V OV 
bys =(o—o Jor +(r —T ‘ey+(u —v Na+ (EE ane + (se )aay + (are te 8x, (S) 
in which 
in which 
~ sick z ee )=* i ==" (1) 
