on Change of Temperature. 419 
and 
V y(l + q + X) x^ + q ~ p) y /A . „f 
-i(i+q-p)-r-2 
£ x(l+p + \)+fy 1+f ij 
XIII. To find the position at the first temperature of a line 
for which the position at the second temperature is given, the 
above formulae must be reversed. We then get, still neglect- 
ing small quantities of the second order, 
|=(l-p + *)f-/, 
y v v 
and (for use when 97 is small), 
~=(i-»-+i>)|+/§f-d|-*, 
I =(!-*+,) f+f 
XIV, The equation to the ellipsoid at the second tempe- 
rature being 
it follows that the radius vector OP / ( = p) having direction- 
cosines /, to, n is given by the equation 
\ = l-2\-2(pP + qm 2 + rn 2 + dm + enl +flm), 
P 
whence 
p — l=\+pl 2 + qm 2 + rn 2 + dmn + en I +flm . 
The absolute coefficient of expansion for the crystal-line having 
the direction I to n is therefore given by the right-hand side of 
this equation, in which A, represents the absolute coefficient of 
expansion of the crystal-line OB. 
If the absolute coefficient of expansion of a given line of the 
crystal be determined by any method, the unknown quantity 
\ in the above formula can be eliminated. 
XV. We now proceed to apply this method to the calcula- 
tion of the thermic axes of anorthite. 
The following Table gives the parameters, as calculated by 
Beckenkamp from the observed angles, and also the logarithms 
of the various terms required in the calculation ofp, q, r, d, e,f. 
2G2 
