on Change of Temperature. 349 
Substituting these values in the above formula we get 
a! c' 
8 S- a c 
. . (IX.) 
1 sin /3 sin (2^+|S)' 
Since «' ^ 
v— /3 , , a c 
^= 2 andtan %= / 3'_ / 3' 
the equation (IX.) may be expressed also in 
the following 
ways 
of d 
8 8- a a ? - .£'-* ■ 
• • (X.) 
(b) From the above we also have 
A-8=(8 1 -8)cos 2 (^ + ^), 
and 
where A and A x are the expansions along OA, OC. 
If 8 2 be the expansion perpendicular to the symmetry-plane. 
rV 
A' 
a' 
8 2 = 
B" 
-1 = 
= \ = 
= A ~ 
a ' 
= 1 + A 
similarly 
8 2 = 
=Ai- 
d-c 
c 
Hence 
8 2 -8=- a -^ + (8 1 -8)cos 2 (f + ^ 
= -^+(8,-8)00^^; 
from either of which equations 8 2 — 8 can be found. 
(10) We shall illustrate the above formulae by application to 
the same case as before. 
From the angles given on page 294, it follows from the 
usual formulae, namely, 
tan ma 
a= — , 
sin ac 
_ cot lb 
sin da ' 
/3=180°-ac, 
