on Change of Temperature. 345 
of the lines OA and OC the corresponding alterations of a 
line OX in the plane AOC and initially at right angles to OC 
(fig. 8). 
Denote as usual the angle AOC by ft. 
(1) Let a, z be the coordinates of a point P referred to the 
rectangular axes OX, OC, and w x , z x the coordinates of the 
same point referred to the oblique axes OA, OC; draw PN 
parallel to CO and meeting OX, OA in N and M respectively. 
Then # = ON, z = ~PN, 
fl?! = OM, 2x = PM; 
whence 
x = x\ sin 3, z = z x + x x cos 3, "I 
*«}•■• (L) 
and x-,=-r—„, z-,=z — x cot 3. 
sm/3' J 
(2) At the second temperature OA, OC become OA', OC 7 , 
whilst 3 becomes 3' and OP, OP'; if, as before, a line P'M' 
be drawn parallel to OC' or OC, then, since parallel lines or 
parts of the same line retain their ratios on change of tempera- 
ture of the crystal, 
OM'_OA'_A^ 
OM ~OA ~A' 
P'M' _ PC _ U. 
PM ~ OC ~ C ' 
whence, if f 1} £i be the coordinates OM', P'M', of P' referred to 
the axes OA', OC, 
(3) If, further, £, £ be the coordinates of P' referred to the 
old rectangular axes OX, OC, then, just as in equations (I.), 
t=k'mxp, 
?=& + &<*» ?; 
whence the rectangular coordinates of P', namely f, £, are 
related to the rectangular coordinates of P, namely x } z, by 
the equations 
A' sin/3' 
A sin r 
