ON DOUBLE REFRACTION, 281 
elementary plane passing through the point («, y, z) in such a direction that 
in the undisturbed state of the medium the same plane of particles was 
perpendicular to the axis of wv, dS denoting the actual area of the element. 
Consider the equilibrium of an elementary tetrahedron of the medium, the 
sides of which are perpendicular to the axes of w, y, z, and the base in the 
direction of a plane which was perpendicular to the axis of w; and let 
1, m, n be the direction-cosines of the base; then 
P'i.=lA+mF+n£, T= F+mB+nD, T’,,=/E+mD+nC; (34) 
but to the first order of small quantities 
= r= — ay P 
dy’ dz’ 
substituting in (34), and writing down the other corresponding equations, 
we have 
P =A . =n) 7 po? ig See et) 
l=1, m=— 
dz dz dx dx dy 
Pp’ =B-DW_r dv T Las Ae _ pw T_=-E- pm C du | 
dz da — dx dy dy dz (35) 
dw dw du du dv dv 
sta BW ca Foal ae pe) ee eae a le QE) ee ey) Ves 
dx dy T, cay dz ye ow aS 
Lastly, since an ca aya area dS originally perpendicular to the axis of x 
becomes by extension ( 1 Ae ta) dS, and similarly with regard to y and z, 
P.=C—E 
we have 
dw 
P.:P = hie! Raine — UN Dab 2 
x sey se Dy Th. Ta. 1404 
ee an eae ae es ee ee 
ie i em + « « (86) 
lu 
i aE ey age =T,,: T= aul 
z z P24 ze zy 1 ty dx + 
Expressing P_, T,,, &c. in terms of P’,, T",» &c. by (36), then P’, ii &e. 
in terms of A, B, C, D, E, F by (85), and lastly substituting for i. Ls Cree 
the expressions given by Cauchy, we find 
) 
P= a (1 +9,) +o + eS 
P= (145 gees 
dx 
P=€(14+7)+e7 BS 
T,=B(1+2 +P Ete Sl e=B(1+5 ates “+E o 
dz 
(37) 
T ik 4B mre Gt =e(1 +E) ee 4q q Ww 
dz 
+e S4a % jg Te (1+ + a 
a 
‘a y 
