774 
DE. PLUCKEE ON A NEW GEOMETER OE SPACE. 
plex assumes the form of the equation (50); the form of the tWo equations being the 
same as in the general case, where the direction of the third axis is obliqtie to xy. 
If in the case of uniaxal crystals the circular section of E is congrilelit with the sec- 
tion xy of the crystal, we get in order to represent the complex of double refracted 
rays, on putting a=0, the following equation, 
indicating that the plane of refraction is congruent with the plane of incidence, or, in 
other terms, that both the' ordinary and the extraordinary ray into which any incident 
ray, starting from OZ, is divided by double refraction, likewise meet OZ. 
34. The preceding fragmentary researches on double refraction — only calculated to 
present a new and curious instance of a complex — may be concluded by a last remark. 
All the results we have hitherto obtained, especially the determination of the com- 
plex of double refracted rays, only depend, 1st, upon the direction of the diameter of the 
ellipsoid E conjugate to the section of the crystal; 2ndly, upon the ratio of the axes of 
the elliptical trace along which the same ellipsoid meets that section. Here, therefore, 
the third auxiliary ellipsoid E, 21721 2 7, 
ax oy ~\“Cz — aoc^ 
may be replaced by the following one, 
ax 2 -{-hy 2 -\-c'z 2 =l, 
which is similar to it. It is immediately seen that, along the different directions, the 
reciprocal values of optical elasticity within the crystal are indicated by the radii vec- 
tor es of the new ellipsoid, as the squares of these values are represented by the radii 
vectores of the second auxiliary ellipsoid, 
fflV + b 2 y 2 + c V = 1 . 
Additional Note. 
Deceived December 11, 1865. 
I. Coordinates of a right line. 
1. A right line, if considered as an axis round which a plane revolves, is determined 
by any two positions of the revolving plane ; analytically, by means of two groups of 
plane-coordinates. If considered as a geometrical locus, described by a point, it is 
determined by any two positions of the moving point ; analytically, by means of two 
groups of point-coordinates. 
Let the plane- and point-coordinates 
ST '57 ST 
t U V 
— , —5 —5 
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