68 Professor Hamittron’s Third Supplement 
u, » being functions of the colour y, of which » involves also the final co-ordinates, 
and ,' the initial co-ordinates, when the extreme media are atmospheres: and then the 
equations (C°) reduce themselves at once to the following expressions of the form 
da ae ov + ae oy + ee éz + ae ov + =e oy + = é Ne } 
Aye) =" (rs Sr + oe oy + oz + ae ou’ + ae oy + os 8x) 3 
éa = sae ou + aay ¥ = oe oz’ + a ou + a sy + a 8x)» 
so= 2 sal ay ou + a4 oy se z+ = ou + - gy + = 8x) + 
In general we see that the twenty-four coefficients of the expressions (D*) can easily 
be deduced, by (C°), from the partial differentials of the two first orders of the cha- 
racteristic function 7, and of the extreme medium-functions v, v': we have for ex- 
ample 
Oni Wl oe Se _ & je Oh (aa _ oe ) 
Seo” Ss \3e  Sade/ ov” Sad \oxdy Soe 
8a a ea ov ee Sv (a _&e ) 
dy v 8B \evdy sady/ vw” SadB\dy2 —- day 
(G’) 
Spa es a Sake Va On 
eet a ay spas) a dae[3 Oz" ee) 
i ou 1 
uo” a — 5Bay) ~ See 3a8 ap? 
iN 
Ox 
vw 
v' having the same ee as in the tenth number. The same twenty-four coefh- 
cients of (D") may also be deduced (as we have said) from the partial differentials of 
the two first orders of the other related and auxiliary functions: or eyen from the 
partial differentials of the three first orders of the characteristic function VY” alone. 
Let us therefore suppose that these twenty-four coefficients of the expressions (D") 
are known, and let us consider their geometrical meanings and uses: that is, their 
connexions with questions respecting the infinitely small variations of the extreme 
directions or tangents of a luminous path, arising from variations of the extreme 
points and of the colour. 
In discussing these connexions, it is evidently permitted, by the linear form of the 
differential expressions (D°), to consider separately and successively the influence of 
the seven variations 6x, dy, dz, a’, dy’, d2', dy, of the extreme co-ordinates and the 
colour, or the influence of any groupes of these seven variations, on the four varia- 
tions da, 63, da’, o', of the extreme small cosines of direction. Thus, if it be required 
