8 Professor Hamitton’s Third Supplement 
straight, and in its whole extent we have not only the equations (J¥”) but also the 
following, 
1 C) ov 
= const. , las const. , ee const. , (Z) 
the constants being by (B) those functions of the final direction-cosines and of the 
colour which we have denoted by 
cv dv ov 
ga’? 88” by’ 
and which are here independent of the co-ordinates. In general, if we consider the 
final co-ordinates and the colour as constant, the relations (Z) between the initial co- 
ordinates are forms for the equations of a ray. And though we have hitherto consi- 
dered rectangular co-ordinates only, yet we shall show in a future number that there 
are analogous results for oblique and even for polar co-ordinates. 
Transformations of the Fundamental Formula. New View of the Auxiliary 
Function WV ; New Auxiliary Function T. Deductions of the Characteristic 
and Auxiliary Functions, V, W, T, each from each. General Theorem of 
Maxima and Minima, which includes all the details of such deductions. Remarks 
on the respective advantages of the Characteristic and Auxiliary Functions. 
4. The fundamental equation (7) may be put under the form 
v 
SV = ob4 — oéu' + roy — roy + vog— vee + = 8x3 (A ) 
employing the definitions (#7), and introducing the variation of colour ; it admits 
also of the two following general transformations, 
V 
SW = ado + ydr + xu + o'da! + 78y' +82" ~~ 8x, (B’) 
and 
, , , , ne OV U 
8 T= x80 — x80 + yor — or + zdv —2'bv' — =~ oy, (C) 
XxX 
in which 
W=—-—V +40 + Yt + Zu, (D’) 
and 
T=W—a's -y'7 — 2. (E’) 
In the two foregoing Supplements, the quantity 7” was introduced, and was consi- 
dered as a function of the final direction-cosines a, B, y, the final medium being 
regarded as uniform, and the luminous origin and colour as given ; we shall now take 
another and a more general view of this auxiliary function 7”, and shall consider it 
