DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
761 
unit as w. If both systems of axes are congruent, the wave-surface is represented by 
the well-kndwn equation 
(«V+% 2 +cVX^+^+^)-[« 2 (^+c> 2 +^« 2 + c2 )/+^K+ J2 K]+ aW = 0 ’ • ( 6 ) 
which, for simplicity, may be written thus, 
0 = 0 . 
4. The wave-surface is intimately connected with three ellipsoids, the equations of 
which are 2 2 2 
^2 +£2 = 1 j (?) 
«V+5y+cV= 1, (8) 
2^ 
h +£ +r* =1 ( 9 ) 
By means of the first and the second ellipsoid the wave-surface may be obtained most 
easily. The third ellipsoid has been introduced by myself on account of the following 
remarkable property. With regard to this ellipsoid the wave-surface is its own polar 
surface, i. e. the polar plane of any point of the surface touches it in another point, and 
vice versd, the pole of any plane tangent to the surface is one of its points. 
The wave-surface and the three ellipsoids depend upon the same constants. When 
the crystal turns around the point of incidence O, both the surface and the three ellip- 
soids simultaneously turn with it. In the new position their equations involve three 
new constants, indicating the position of the axes of elasticity with regard to the axes 
of coordinates. Now the wave-surface may be represented by 
O'=0, 
and the third ellipsoid in the corresponding position by 
A# 2 +B#;y+Oy 2 d-2D#z+2%;s+F;s 2 ---l=E=0 (10) 
From the six constants of this equation, which may be regarded as known, you may 
derive the six constants of the wave-surface by determining both the direction and the 
length of the axes of the third ellipsoid. 
Within the plane xy , supposed to be any section whatever of the crystal, OX and 
OY may be directed along the axes of the ellipse along which this plane is intersected 
by the third ellipsoid. Accordingly the constant B disappears from the last equation. 
Besides, if OZ be directed along that diameter of the ellipsoid which is conjugate to 
the plane xy, and cease therefore, in the general case, to be perpendicular to it, both 
constants D and E likewise disappear. 
5. According to Huyghens’s principle, we obtain both rays into which an incident 
ray is divided, when entering the crystal, by the following general construction. Con- 
struct the two planes passing through the trace RR and tangent to the wave-surface 
described within the crystal around the point of incidence O. Let H and H' be the 
