Boorn—On Fresnel’s Wave Surface. 385 
it follows at once that the locus of the foot of the perpendicular 
from the origin on a tangent plane to the wave surface is 
x 3 y : z 
ee a ey so ee a ee 
2 2 w2 
and as the inverse of this latter surface with respect to the sphere 
e+y+s2?—-1=0 
is the reciprocal of the wave surface, its equation is therefore 
e oP 3 
=e Se ee 
O(e?+yt+s)-1l Petty +2)-1 - C(Pe+yt+es)-l ’ 
but this is the wave surface of the reciprocal of the ellipsoid of 
elasticity. In other words, “the reciprocal of the wave surface is 
the wave surface of the reciprocal of the ellipsoid of elasticity.’’? 
Now, the equation of the wave surface for the ellipsoid 
eo yf # 5 
e a B ae e —1=0 
must be, by the foregoing, 
T? + vi vev3v,= 9, 
where 7' stands for 
eC (a + 0) + 2a°ey*? + va (0? +0) -(v? +e), 
and v, for 
co / &—0+2a/P—C+f/a—e, 
with corresponding values for v2v3»,; hence the equation in tan- 
gential coordinates of the wave surface of 
aa? + by? + cs’ -1=0 
is 
w+ AAAs, = 0,7 
where a stands for 
Ne? (a? + 0) + 2Ware + va’ (0? +) — (+e), 
and X, for er 
Ne. /@—-B+ va /P—e+ fa —e, 
with corresponding values for 2, As, As. This mode of writing its 
equation shows that 
M9, opel. A 0s A= 0 
1 See Salmon’s Swifaces, page 426. 
