348 Mr. L. Fletcher on the Dilatation of Crystals 
(8) Substituting these values of P, R, E in the expression 
for the inclinations of the thermic axes, we get 
tan 2ty = 
E 
~P-R 
P 
e 
— r 
ft' cos /3' cf cos/3 
ft sin c sin /3 
ft' sin /S' c' 
« sin /3 c 
from which, it m: 
fying, we get 
ay be re; 
tan 2-v^- 
mar 
«' 
ked, 
J 
c 
\ has disappeared. 
- tan/3 
& 
-/3 
Simpli- 
ft'_c' 
ft c 
/3'-/3 
If ft' c' 
tan /3 + 1 
ft c 
= tan % , we have ^= 2t_l. . . . (VIII.) 
These simple formulae (VIII.) thus give the directions of the 
thermic axes in terms of the parameters at the two tempe- 
ratures. 
(9) To determine the principal expansions directly from 
the parameters at the two temperatures. 
(ft) From Prop. XIV., p. 292, we know that if A andA x 
be the expansions in directions inclined to a thermic axis at 
angles 6 and </> respectively, 
A-S=(8 l -S) sin 2 0, 
A 1 -8 = (8 1 -S)sin 2 (/>; 
whence 
• A= (S!-S) (sin 2 <j>- sin 2 6) 
= (h x -h) sin (<f> + 6) sin (<f> - 6), 
and g _g A,— A 
sin (0 + 0) sin(</> — 6)' 
If the directions 6, <f> coincide with the lines OA, OC respec- 
tively, 
and ^-A=~-^ 
(J A 
_ c' ft' 
c a* 
