On Systems of Rays. 127 
to the time 7 + AV, be upon the ray that passes through the point w y z of the first 
plane wave, it will be also on all the other infinitely near plane waves which corres- 
pond to the same time V +AV, these other waves having passed through the point 
wy z at the time VY, and having made infinitely small angles with the first plane 
wave ; we are therefore to find the co-ordinaets # + Az, y + Ay, = + Az, of the second 
point upon the ray, by seeking the intersection of the second wave (C’) with all those 
other waves which are obtained from it by assigning to o, 7, v, any infinitely small 
variations consistent with the relation 
0= 50 ie oo + = oT Tae vu; 
and thus we find 
a_Av _80 B _Ay _ 80 dies Az _6Q (D") 
v AV 8’ v AV &’? v AV wv’ 
as in the second number of this Supplement, and therefore 
v =ac +PBr +y, 
O = ado + Ber + you, 
dv = oda +708 + vdy, 
and finally 
ov ov ov 
ia Oy Fc a Ty; Sina (E*) 
if we denote by v the reciprocal of the undulatory velocity with which the light is 
propagated along the ray, and by a, , y, the cosines of the angles which the ray makes 
with the axes of co-ordinates. We see, therefore, by the foregoing reasoning, which 
it is easy to extend to the case of curved waves and of variable media, that the com- 
ponents c,7, v, of normal slowness of a wave, or the partial differential coefficients of 
the first order of the time- pene V, are equal to the partial differential coefficients 
dv dv 
of the first order, = aoe yy? of the undulatory slowness v of propagation along 
the ray, when this latter sowineks is expressed as a homogeneous function of the 
first dimension of the direction-cosines a B y of the ray : which is the general theorem 
of mathematical optics, expressed by our fundamental formula (4). 
That general theorem does not appear to have been perceived by other writers ; 
nor do they seem to have distinctly thought of the components of normal slowness, 
nor of the function of which these components are partial differential coefficients, that 
is, the time V of propagation of light from one variable point to another, through 
any combination of uniform or variable media, considered as depending on the final 
and initial co-ordinates and on the colour: much less do those who have hitherto 
written upon light, appear to have thought of this time-function V as a CHARACTER- 
