on Change of Temperature. 283 
and thus to the same approximation, 
Q"? // -QP = (d-e)(m-n-d + e) sm |Q F , . . (7) 
Substituting the above values of m, n } d, e, it will be found in 
this manner that 
Q"P"- QP= 0-000000012384 sin 2QP, . . (8) 
which is virtually the same result as before. 
Again, from equation (1) we have for the intermediate 
temperature t 3 , 
tanP"X=?tanPX, 

Whence ftanPX-tanPX 
tan FT-ta (P"X-PX) = f +tanP , XtmPg 
and, neglecting small quantities of the second order, 
P // p= ( a - & ) sin2 PX (9) 
If for the sake of example we take PX=30°, we find in this 
way that the displacement of P for the change of temperature 
10° — 50° C. is about 23 // : the change in the inclination of P 
to Q for the change of temperature 10° — 50° is thus less than 
the -99V0 P ar *- °f the absolute displacements of the poles P and Q. 
From the above we think it will be clear that lines may be 
isotropic for one pair of temperatures without being so for 
others; that when there are two perpendicular atropic lines, 
this variation of angle will be a small quantity of the second 
order ; and, further, that lines may be isotropic for one pair 
of temperatures and yet be absolutely without claim to im- 
portance as axes of the crystal. 
In the previous paper it was mentioned that, whether the 
indices of the planes be rational or irrational, or, what comes 
to the same thing, whether the planes be natural or artificial, 
the indices will be unaltered by any change of temperature 
of the crystals ; and, further, that the anharmonic ratios of 
any four natural or artificial planes in the same zone must be 
likewise constant on change of temperature, for they can be 
expressed in terms of the constant indices*. Considerable use 
of the latter property is made in the proof of the various pro- 
positions given in the present paper ; as in the case of the 
anharmonic ratios themselves, the formulas are true whatever 
* 'A Tract on Crystallography,' by W. H. Miller, 1863, p. 10. 
