286 FRESNEL ON DOUBLE REFRACTION, 



The small displacements imrallel to the axes of any diametral 

 section lohatever of the surface of elasticity, do not tend to 

 separate the molecules of the succeeding strata from the normal 

 plane drawn through their direction. 



I shall now demonstrate that the greatest and least radius 

 vector, or the two axes of the diametral section, possess the pro- 

 perty just announced; that is to say, the displacements along 

 each of these two axes excite elastic forces, the component of 

 which perpendicular to their direction is found at the same time 

 perpendicular to the plane of the diametral section. 



In fact, let 07 = B y + C 2" be the equation of the cutting plane 

 passing through the centre of the surface of elasticity ; the equa- 

 tion of condition which expresses that this plane contains the 

 radius vector, whose inclinations to the axes of x, y, z are respec- 

 tively X, Yj Z, is 



cos X = B . cos Y + C . cos Z. 



We have, besides, between the angles X, Y, Z, the relation 



cos^X + cos^ Y + cos^Z = 1 ; 

 and for the equation of the surface of elasticity, 



u^ = c2 cos^ X -f 4^ cos^ Y + c2 . cos^ Z. 



The radius vector (y) attains its maximum or its minimum when 

 its differential becomes nothing ; we have therefore in this case, 

 differentiating the equation of the surface with respect to the 

 angle X, 



= a-.cosX sinX + i^.cos Ysin Y. -7^ fc^.cosZ.sinZ. -r^. 



a A a X 



If we differentiate similarly the two preceding equations, we have 



cos X sin X + cos Y sin Y . -^^ + cos Z . sin Z . -yw = 0, 

 « X a X 



— sinX + Bsin Y. -j^ +C.feinZ.;Tv = 0; 

 ct \. a A. 



^ Y dTi 

 whence we obtain for -pj^r and -7^^ the foUowino- values : 

 dX dX " 



dY _ sin X (C cos X + cos Z) 

 dX~ sin Y (B cos Z - C cos Y)' 



and dZ _ — sin X (B cos X -f- cos Y) 



^X "" "sm"Z"(B cos^Z^ C .cos Y) ' 



