20 Professor Hamitton’s Third Supplement 
comparing the equations which connect the three functions themselves: that is, by dif- 
ferentiating and comparing the three forms (4') (B’) (C’) of the fundamental equa- 
tion (4), and the equations into which these forms (4’) (B) (C’) resolve themselves. 
Thus, to deduce the twenty-eight partial differential coefficients of the second 
order, of the characteristic function 7, taken with respect to the extreme co-ordinates 
and the colour, from the coefficients of the same order of the auxiliary function /V, 
or 7, we are to differentiate the equations into which (B’) or (C’) resolves itself, 
together with the relations between the variables on which JV or 7’ depends, if any 
such relations exist; and then by elimination to deduce the variations of the first 
order of the seven coefficients of the variation (4’) as linear functions of the seven 
variations of the first order of the extreme co-ordinates and the colour: these seven 
linear functions will have forty-nine coefficients, of which, however, only twenty- 
eight will be distinct, and these will be the coefficients sought. 
More particularly, if the final medium be variable, and if it be required to deduce 
the coefficients of the second order of V from those of V7, we first obtain expres- 
sions for 8s, 87, dv, as linear functions of 8, dy, dz, 82’, dy’, 62’, dy, from the differen- 
tials of the three first equations (H’), deduced from (B’), expressions which will 
necessarily satisfy the first condition (H7); we then substitute these expressions for 
éc, ér, dv, in the differentials of the three equations (G’), deduced from (B'), so as to 
get analogous expressions for 80’, or, 6v, which must satisfy the second condition 
(#7); and substituting the same expressions for 6s, dr, dv, in the differential of the 
last equation (/7’), also deduced from (B’), we get an expression of the same kind for 
ee 5 - : : 
6 wa after which, we have only to compare the expressions so obtained, with the 
following, that is, with the differentials of the equations into which the formula (4’) 
has ats 
8 OF aig eee spp ng ae ety oO aati 
oe Sedy Y * Sade" Trae” TB a YT arae * * 323, °% 
a ap SV oo 1 piORLA 
or= sae ¥ le ae ‘aa aa La may + Bay OX? 
eV i“ 4 &V 
du= ane + oy + Y 32+ + oT a0! ‘aa + —— 62’ + yay 8X 
ee es a ic V eR ed o? Ma : = 
om, 27 ev 52 V eV SV, ‘e 
ae Say aoe &Y + 37 & tara +o =u nia Bioyeeat 
ee cake eV eV S°V SV, yo*7 
—w =a es 7 oy + ser ae @ +y797 et vay +o gor + 37 by. 
4 o°2V 2! SELL eee gL 
eh ee? om y+ be + pte + aa Y by + ay oz tea 8x 
