on Change of Temperature. 421 
The values of p, g, r, d, e,f deduced from these by the for- 
mulae of § VII. are as follows: — ■ 
20°— 80° C. 
20°— 140° C. 
20°— 200° C. 
p= +0-0004442 
+0-0009482 
+00017017 
q= -0-0000010 
-0-0000025 
-0-0000238 
r= -0-0000855 
-0-0000037 
+0-0000327 
d= +0-0000419 
+0-0000888 
+0-0001257 
e= +0-0001522 
-I-00004701 
+0-0010007 
/= -0-0000376 
-0-0000706 
-0-0004525 
XVI. We may in the first place remark that the expansions 
p, q, r and the angles of rotation d, e, f are far from propor- 
tional to the changes of temperature ; / in fact is greater at 
140° C. than at 80° C. and 200° C. 
The thermic axes for the different pairs of temperatures are 
the axes of the following quadrics (§ XI.) : — 
20°— 80° 53(k 2 + Uif + 42ijz + lh2zas- 38^ = 1, 
20°- 
20°- 
•140 c 
952^ 2 + ?/ 2 + 89ijz+ 470^' 
-200° 16(]9^ 2 -57 ? / + 12Qyz + lOOlzas- 
-452^ = 1. 
It will be found that the values of A/ &c. used by Becken- 
kamp (p. 278) correspond to the following quadrics: — 
20°— 80° 529^ 2 + 84y 2 + 42^+ 152^- 37#y = l, 
20°— 140° 952^ + if + 88yz+ 467 zx- 70«y=I, 
20°— 200° 1668^ 2 -57?/ 2 + 126^ + 1000^-451^ = 1; 
and these are virtually the same as the equations just given. 
XVII. Owing to this identity of the equations as calculated 
by these different methods, it was at first intended to assume the 
accuracy of the remainder of Beckenkamp's calculations of the 
axes of the 20° — 200° C. quadric, and to calculate therefrom the 
variation in the angles between these lines at 80° C. and 140° C. 
It was found, however, that from some cause or other the 
angles between the lines given by him are not quite right 
angles; and in fact it will be found that if L'", M'", W be the 
positions at 200° C. of the thermic axes for the pair of tempera- 
tures 20°— 200° C, the logarithms of cos L'"X, cos L"% 
&c. given by Beckenkamp lead to the angles 90° — 16"*87, 
