418 Mr. L. Fletcher on the Dilatation of Crystals 
its initial plane YOZ, and e the angle of displacement of the 
line OX' from its initial position OX. 
We may here conveniently point out that the above proves the 
truth of the remark made on page 271) — that if squares of small 
quantities be neglected, the equations for the determination 
of the thermic axes will be identical whether the lower or the 
higher temperature be the starting-point. In fact, if we pro- 
ceed from the higher to the lower temperature, each one of the 
coefficients p, q, r, d, e,/retains its numerical value but changes 
its sign, and the same quadric can be used for the determina- 
tion of the axes. But as this quadric always gives the positions 
of the axes for the final temperature, it will give in this case 
the position, not at the higher, but at the lower temperature. 
XI. Whatever be the value of k, the principal axes of the 
series of quadrics 
k(x 2 + y 2 + z 2 ) +px 2 + qy 2 + rz 2 + dyz + ezx +fxy = 1 
have the same directions; we may thus diminish the labour 
of numerical calculation by making k + r vanish, and finding 
the axes of the coaxal quadric 
( p — r) x 1 + (q— r)y 2 + dyz + ezx +fxy = 1 . 
By comparison of numerical results the equations given by 
Neumann for the determination of the thermic axes are found 
to refer to this particular quadric. 
XII. Without X being known, the relative angular dis- 
placement of a given line OP can be determined. 
From the above equations (§ IX.) we can write 
|_ 41+P + X)+,/y , 
v ~ y(l + q + X) -y {i+P 9)+f > 
and 
? z(l+r + X) + ex + dy z„ x , 
= /i , , %\ — St = -(l+r-q) + e +d, 
V y(l + q + \) y" u y 
from which cos P'X = — . a — &c. can be calculated. 
When y is zero or very small, we must use the corresponding 
equations wherein y does not appear as a denominator ; 
namely, 
y /i, n\, ,7 -(1 + r— p)+d^ +e 
g _ z(l+r + X)+ex + dy _ x v rj ,r 
f~ X {\ + p + \)+fy „ 
"* X 
= £(1-H— p)-/ i+(/' /y +e, 
