126 Professor Hamitton’s Third Supplement 
we may therefore call the quantities o, r, v, the components of normal slowness, 
because they are equal to the reciprocal of the normal velocity, that is, to the normal 
slowness, multiplied respectively by the direction-cosines of the normal, that is, by the 
cosines of the angles which it makes with the rectangular axes of co-ordinates. 
Such then may be said to be the optical meaning of our quantities o, 7, v, in the 
theory of the propagation of light by waves. And we might easily deduce from this 
meaning, and from the first principles of the undulatory theory, the general expres- 
sion (4) for the variation of the characteristic function 7, which has been proposed 
in the present and former memoirs, as fundamental in mathematical optics. For it is 
an immediate consequence of the dynamical ideas of the undulatory theory of light, 
that for a plane wave of a given direction and colour, in a given uniform medium, the 
normal velocity of propagation is determined, or at least restricted to a finite variety 
of values ; so that this normal velocity may be considered as a function of its cosines 
of direction, volving also the colour, and depending for its form on the nature of 
‘the uniform medium, and on the positions of the axes of co-ordinates, to which the 
angles of direction are referred: and if the medium be variable instead of uniform, 
and the wave curved instead of plane, we must suppose that the normal velocity w is 
still a function of its direction-cosines o(6* +77 +v")~4, r(o +72 +u")-4, v(o? +72 +u")-4, 
and of the colour x, involving also, in this more general case, the co-ordinates &, y, z. 
In this manner we are conducted, by the principles of the undulatory theory, to a 
relation between o, r, v, 2, y, 2, x, of the kind already often employed in the present 
Supplement, namely, 
0=Q=(6 +72 +v)to—1, (M) 
Q+1 being a homogeneous function of «, 7, v, of the first dimension, which satisfies 
therefore the condition 
8Q 0G 6Q 
SriGee ae wo Hl 
ig, hoe SO 
and which involves also in general the co-ordinates 2, y, z, and the colour y, and 
depends for its form on the optical properties of the medium in which the pot x y 2 
is placed. To connect now, for any given point and colour, the velocity and direction 
of the ray with the direction of the normal of the wave, we may suppose, at first, that 
the medium is uniform, and that the wave is plane. The two positions of this plane 
wave, at the time J’, and at the time V+ AV, may be denoted by the equations 
om 
First ot +ty+uz=V + IV, (Cc) 
Second cAaw+ TAy +vAz= AP, 
in which o, 7, v, JV, are constants; and by the principles of the same undulatory 
theory, if the point +A, ¥+Ay, 2+ Az, on the second plane wave, corresponding 
