on Change of Temperature. 299 
of the arc traced out by T must be distant exactly 45° from 
the point of bisection of the arc act. 
To find the change at the second temperature in the mutual 
inclination of two planes which at the first temperature are per- 
pendicular to each other but inclined to the thermic cures at an 
angle of 1° (Prop. VI.). 
If P, Q be the two planes, 
Pa = 36° 46' 32"-58, Qa = Pa + 90°. 
(a) For the motion of P, 
d^-odS^-lS sin 36° 46' 32"-58 sin 55° 18' 10"-88 
= -294"-438, 
tan 2 6 X cot Ta= +0-56030; 
whence the motion of P is — 293 //, 877. 
(b) For the motion of Q, 
1 =-598 // -18 sin 126° 46' 32"-58 sin 145° 18' 10"-88 
= -2 72"- 740, 
tan 2 6 X cot Qa= -0"'26955 ; 
whence the motion of Q is 273 //- 010; and the change of incli- 
nation of P to Q is only 20"-867. 
If, then, we wish to find by trial a pair of planes which can 
be proved by direct measurement of the angle between them 
to retain their perpendicularity, a possible error of one second 
in the measurement of the angle between the planes will lead 
to a possible error of ^q.^- , or 172'52 seconds in the deter- 
mination of the position of the pair of planes relative to the 
given plane a. 
To find the motion relative to the plane a of a plane inclined 
to a. at an initial angle of 1° (Prop. VI.). 
Here# l =-598"-18 sinl sin 19° 31' 38"-3 
= — 3"-4895, 
and 
tan 2 ^ cot Pa = 0: 
whence the motion relative to a is only — 3 //- 4895. Hence 
if we are given the plane a, and are required to find a plane a 
which can be proved by direct measurement to be equally 
inclined to a at the two temperatures, a possible error of one 
second in the measurement of the angle between the planes 
