On Systems of Rays. 131 
It is easy to see that the value of v thus determined is the normal slowness, or reci- 
procal of w, because the expressions (P") give, by (L**), 
os +77 ze v= vy? 5 (Chey 
and since the same expressions give also evidently, by (Q"*), 
3 7 : 
v 
a PS Lae Eee 
0, (S'*) 
we easily deduce the law (G@") of dependence of the normal velocity on the normal 
direction, from the form of FREsNEL’s wave, as we had deduced the latter from the 
former. 
The equations (L"*) (MM) which gave us the equation of the wave in rectangular 
co-ordinates, give also the following polar equation for the reciprocal of its radius- 
vector, that is, for the slowness v of the ray, 
O=v'—v* fab + 6%) + B(e*# +0") + y7(a?* + 8} 
+ (a” + ice J yY) (a°b~ c + [Care= an ae y a- b=); Cr) 
and therefore the following double expression for the square of this slowness, 
veh +o) (@+f+7) 
ee ee 18 
+$(c%-a){ 4A" +t V0 +R +y—A*? Vat+fPh't+y —A}, (U") 
if we put for abridgment 
rae aad 
oes Sh Ap ae 
= eas a pee t= : 
ca 
ee 
supposing therefore a’ > 0’ > c’, the polar equation of Fe wave may be put under 
the form 
ep =h(c* +a) +h (c?—@~) cos. ( (99) + (ye") ), cw") 
e being the radius-vector or velocity, and (p9') (ep") being the angles which this radius 
e makes with two constant radii o', p”, determined by the following cosines of their 
inclinations to the semiaxes of w y z, or of abc, 
‘ 7] (je , aoa’ 18 
Pe a= eas Bip p— Os, Oc — =P c= / ae . (X") 
The expression (/7"*), for the reciprocal of the square of the re of a ray, has 
been assigned by Fresnex, who has also remarked that it gives always two unequal 
velocities unless the direction p of the ray coincide with some one of the four direc- 
tions +9, +o”, which are opposite two by two, and situated in the plane a c of the 
