on Change of Temperature. 417 
IX. Let the coordinates of any point P at the first tempera- 
ture referred to OX, OY, OZ be os,y,z; the coordinates a/, y', z ! 
of P', the position of P at the second temperature, referred to 
the three oblique axes OX', OY', OZ', will be 
(l+i?+\>, (l + q + \)y, (l+r+\)z. 
But, from the formulae stated in § II., if £, n, £ be the coordi- 
nates of P' referred to the rectangular axes OX, OY, OZ, we 
have 
^=\ 1 x / + \ 2 y / + \ 3 z / , 
7) = ti 1 a f + Wj f + f x, s z f , 
% = v 1 x' + v 2 y' + v z z f . 
Substituting in these formulae the above values of a/, y r i z', 
^15 i*ij v i; & c -> we get (neglecting squares of small quantities) 
p= (l+ p + X) x+ f(l + q + x)y=x + { P + X)x+fy, 
Y= (l + q + \)y = y + (q + \)y, 
l£=e(l+p + ~K)% + d(l+q + X)y + (l+r + X)z=z + ex + dy 
and, reversing the formulae, 
y—r)—{q + X)7), 
z=£-e%-d V -(r+\)£ 
X. If all the points P lie on a sphere of unit radius having 
the equation x?+y 2 + z 2 =l, all the points P' will lie on an 
ellipsoid having the following equation :— 
[f -2(p+\)? -2M1 + [v 2 -2(q + \)v 2 ] 
+ [?-2eft-2dvZ-2(r + X)?-] = l, 
or 
(l-2M[f + ^ + ? 2 ]-2[p| 2 + ^ 2 + ^ 2 + «+^+/^]=l. 
The thermic axes, being the axes of this ellipsoid, will have the 
same direction as the axes of the quadric 
pa? + qy' 2 + rz 2 + dyz + ezx +fxy = 1, 
all the coefficients of which are known in terms of the para- 
meters, and have, as shown above, extremely simple physical 
representations; p, q, r being respectively differences of the 
expansions of the lines OX, OY, OZ from the expansion 
along OB, d the angle of displacement of OY' from its initial 
plane XOY, / the angle of displacement of the same line from 
Phil. May. S. 5. Vol. 16. No. 102. Dee. 1883. 2 G 
