DE. PLUCKEE ON A NEW GEOMETEY OF SPACE. 
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the vertical projection of OZ', the plane xz! is a principal plane of the ellipsoid E, con- 
taining the two optic axes, and OY the mean axis of the ellipsoid E. 
31. If the plane xy is a principal section of the third auxiliary ellipsoid E (and there- 
fore of all auxiliary ellipsoids), the axis OZ', becoming perpendicular to xy, is congruent 
with OZ. Then the equation of the ellipsoid E, referred to rectangular coordinates, 
becomes 
,;2 w 2 ,2 
i-jjL. i_— = 1 
bc'ac'ab ’ 
and may be written thus, 
ax' 2 -f- by 2 cz 2 = abc. 
Hence the equation of the complex is 
arc=bsg (50) 
If the crystal be turned round OY through an angle %, we get, after replacing x and z 
by 
x cos a, — z sm a, 
x sin k-\-z cos a, 
the following equation of the ellipsoid E, 
(a cos 2 a-\-c sin 2 ot)x 2 -\-by 1 — 2(a— c) sin a cos a . xz-j-(a sin 2 a + c cos 2 u)z 2 =abc. . (51) 
The axes of the elliptic trace within xy being always directed along OY and OX, the 
equation of the complex assumes the form of the equation (32), which, after putting 
E=0 and 
A : C : D = (« cos 2 a — c sin 2 a) : b : — {ci— c) sin a cos a, 
passes into the following one, 
(a cos 2 a — c sin 2 u)rc—bsg—(a—c) sin a cos a . <r=0 (52) 
32. The equations (51) and (52) of the last number belong to the case in which one 
of the three axes of elasticity, OY, falls within the section of the crystal. The two 
remaining axes of elasticity are confined within the plane XZ, where one of them, corre- 
sponding to C, makes with OZ an angle a, this angle being counted towards OX. 
The two equations may be regarded as representing the general case of uniaxal crystals 
cut along any plane whatever. Indeed let OC be the single optic axis making with the 
normal to the section xy of the crystal any angle a. Draw through OC the plane xz 
perpendicular to xy, and OY perpendicular to that plane. The rectangular system of 
coordinates being thus determined, the equations (51) and (52), after having replaced 
c by a, will belong to uniaxal crystals. 
33. If the optic axis of an uniaxal crystal falls within the section xy, the equation of 
the complex, on putting a=±7r, becomes 
crc—as§. 
In the case of uniaxal crystals, each plane passing through the optic axis may be 
regarded as a principal section of the ellipsoid E. Therefore the equation of the com- 
mdccclxv. 5 o 
