298 Mr. L. Fletcher on the Dilatation of Crystals 
The second term thus reaches a maximum value of only 
four-fifths of a second, and only exceeds half a second for 
poles in the small arc extending from 35° to 70° from a. The 
second term might thus, in the case of gypsum at least, be 
neglected in comparison with experimental errors. 
To find the exact positions of the lines of greatest and least 
expansion, assuming the measurements of the angles to be perfect. 
From page 296, 
-J— =&=• 0029001. 
sm aa 
whence 
-=2170-332. 
e 
If T be the principal axis nearest to a, then, from Prop. IX., 
tan (2Ta-aa)= (l + -) tan aa = 2171-332 tanaa; 
whence 
2Ta-aa + 90° = 283-46, 
and 
Ta = 35° 46' 32-58. 
To test the accuracy of the formulas we may calculate the 
motions of T and Tx respectively relative to that of a. 
(a) For the motion of T. 
1= -598"-18 sin 35° 46' 32"-58 sin (35° 46' 32"-58 + 18° 31' 38"-3) 
= -284"-000, 
^-fl^tan^cotTa 
= +0"-54266; 
whence the motion of T to the second order of small quantities 
is 283"-457. 
(//) For the motion of T 1} 
d l= -598-18 sin 125° 46' 32' / -58 sin (125° 46' 32 // -58 
= -283-178, +18° 31' 38-3) 
6 2 -d 1 =tan 2 0! cot T,a= -0-28014; 
whence the motion of T x to the second order of small quanti- 
ties is 283—458. T and T a are therefore isotropic, and distant 
90° from each other at both temperatures. 
We may remark that the values above deduced agree very 
satisfactorily with Prop. IV., which states that the middle point 
