on Change of Temperature. 291 
Q, then (Prop. VI.) 
Q'OQ =k sin Qa sin Qa = k cos Pa cos Pa. 
Now the crystal-line RQ tangent to the circle at Q, and 
thus parallel to the line OP, will become a tangent line r'q' 
to the ellipse and parallel to the tangent to the circle at Q'; 
since parallel lines of the crystal are rotated through the same 
angle, the crystal-line OP will be rotated through the same 
angle as the tangent to the circle at Q, i. e. through the angle 
k cos Pa cos Pa. The difference of motion of OP the normal 
and OP the ray will therefore be 
k sin Pa sin Pa — h cos Pa cos Pa, 
or — k cos (Pa + Pa), 
which will only vanish when 
Pa + Pa=(2w + l)|, 
which is the case when OP is a line of greatest or least ex- 
pansion (see Prop. XI. d). 
Prop. XIII. To find a physical expression for the quantity k 
(fig. 5). 
We have already seen that in the formula 6= k sin Pa sin Pa, 
a, a are two isotropic poles ; we shall now show that the co- 
efficient h when expressed in circular measure is equal to the 
difference of the greatest and least coefficients of expansion 
of the crystal-lines lying in the plane of the zone. 
Let Oa, Oa be the normals to a pair of isotropic planes a, a; 
and let OL, OM, OL', OH' be the positions of the crystal- 
lines of greatest and least expansion at the two temperatures. 
If OP, OP' be the normals to LM, L'M', and 8, S\ the coeffi- 
cients of expansion of the lines OL, OM, we must have 
whence 
0L '_n .*> ow H4-M 
0L- (1 + 8) > OM = ( 1 + S i)> 
tan POL' l + o 
tan POL 1 + oY 
Since by Prop. IX. the position of L, and by Prop. I, the arc 
L'a' can be calculated to any degree of accuracy, and also 
the arc P'a' corresponding to any known arc Pa, the relation 
1 + 8 = tan(P'a'-L / a') 
l + o\~ tan (Pa— La) 
is one from which the ratio = s- can likewise be calculated 
i- + o 1 
to any degree of precision. 
^ Y2 
