162 F. F. Wright — Transmission of Light through 

 9X 1 , f 



-§; = c («„* + «,.l + «i,0 



5Y 1 



-& = -(uJ + «,>V + aJ) (1) 



in which the so-called polarization constants a„ . . . , a s , . . . , 

 a 31 . • . , are simple determinate functions of e„ . . . , e 21 . . . , 

 e si . . . , and c a and for which the relations a hk — a kh hold 

 true just as e hk = e kh . These equations indicate that a function 

 of the second degree is possible whose partial differential 

 quotients with respect to £, w, f, are equal, respectively, to 



3X 3Y «9Z . 



*' 57' 5^' ThlS funCtl0n 1S 



2cG = a„f + a 22 7 ? a + a 33 r + 2rt 33 ^ + 2a 31 ^ + 2o 12 |7 ? = constant. (8) 



From energy considerations it is evident that this equation (8) 

 must represent an ellipsoid ; and if, in it, the constant be 

 2cG=l, the equation is then that of a triaxial ellipsoid referred 

 to a coordinate system of any position. This ellipsoid is the 

 "index ellipsoid" of MacCullagh, or "ellipsoid of elasticity" 

 of Kirchhoff, or the "indicatrix" of Fletcher. The coordi- 

 nate axes can be brought to coincide with the principal 

 ellipsoidal axes by use of the usual transformation equations : 



a'p\ 



+ b*p\ +<?p\ 



= a, 



«y, 



+&V. +c y, 



= «■ 



aV, 



+ b\\ + cV 3 



= «, 



a\j l r l 



+ b 2 qj\ +c*q 3 r 3 



= a 2 



a'rjo. 



+ b\p,-\ c\p s 



= «3 



a*p x q 



l + VP& + °'P& 



= », 



(9) 



in which _#>„ £? 2 , j? 3 , ^„ q„ q^ and r„ r 2 , r 3 , are the direction 

 cosines between the new coordinate axes x, y, z, and the x', y', z' 

 of the old system respectively. Referred to the principal 

 ellipsoidal axes equation (9) becomes a^ + SV + ^f 2 = 1, in 

 which a, o, c, are the principal light velocities of the crystal. 

 The symmetry axes of this index ellipsoid are the reciprocals 

 of a, b, e, or directly the principal refractive indices of the 

 crystal. In geometric problems of reflection and refraction, 

 the index ellipsoid and the index surface derived from it are 

 specially useful. 



