( 64 ) 
ay, A2 ay u? az VI 
eN 
9 ») 6 
ay; vg a, vg"? az v 
—0. . (59) 
This is again a quadratic equation in og", so that for every 
direction of the ray radius there are two velocities of ray. The 
wave-surface consists therefore of 2 sheets, which both enclose 0. 
From these last equations we can easily find the equation of the 
wave-surface in coordinates of points. In order to find those of the 
U 
transformed wave-surface, we have only to substitute =e 
gv 
y' dd je 
Ü=S v=— and ep? = 2? 4+ 9? + 22, 
v ° 
If #, y and z are the coordinates of a point of the wave-surface 
with the principal directions as axes of the coordinates, we find easily : 
OS Ene ln WES Cay ey ce me 
Q 2 ne 
1 Em 6 m Nm 
= MAW ae SS 
W Em Gin Ym ee Owe Ne 
The equation of the wave-surtace becomes then: 
whereas [/ Ur Uy Hz = 
72 y2 Nop y2 22 
: e id X tee a Oe 2 
(EE +5) nlt 45) ete Ees Bn Om tn — 
2 nes 9, 
Ee Se We 
DE her (2% Wm + 7%c on) =e y (n° En + a Hm) + 
je ae 
re z (ee Con = a en) \ ener? Eg CE (60) 
we Ww 
This is the same equation as the equation in tangential coordinates 
except that the quantities €, be, Me, Em Sm; Ym) ¥ and w are replaced 
by their inverse values. The surface must therefore be dualistic 
with itself. From every property which is invariant with respect 
to transformations of affinity, a dualistic one may be deduced, by 
replacing a point by a plane with the same coordinates and vice 
versa, replacing at the same time & Sey Mes Em Sm Ym © and w by 
their reciprocal values. 
The wave-surface is of the 4t degree and of the 4% class with 
the coordinate planes as diametral planes, with the opposite axis 
of coordinates as direction of chords. 
Every line through O intersects the wave-surface on either side 
of O in two real points, while parallel to a definite plane on cither 
side of O two real tangent planes may be drawn. 
