Crystals on Change of Temperature. 95 



ginary, will have the same directions at the two temperatures, 

 their equation being 



[oismOx 2 — (acos# — y cos <j))xz + y sin<£2 2 = 0. . (8) 



We have seen above that in the limiting case these lines are 

 real. Also one pair of crystal-lines, inclined at angles of -^ 



and i/r + — to the axis OX at the initial temperature t, will be 



at right angles at the temperature r f } the angle yjr being given 

 by the very simple relation 



, or 2 sin (0-0) 



tan2ajr= — • 



y a. 



If the lines given by (8) have been atropic during the interval 

 from t to r' , the same crystal-lines cannot have been at right 

 angles during the whole of this interval ; for if the ratios 

 a sin 6 : a cos 6 — y cos (/> : y sin <p be constant during the in- 

 terval, ty can only be constant when 6 and <f> are zero and 

 a — y. Further, it may be shown that if any crystal-line OP 

 at the temperature r makes an angle ty with the axis OX, a 

 second crystal-line OQ making an angle ^ + e with the axis 

 OX can be found which has the same inclination to the crystal- 

 line OP at the two temperatures. If a sin 6 = a, 7 cos <£ = &, 

 a cos 6 — c,y sin (f) = d, and 1/^ be the inclination of the crystal- 

 line OP to the axis OX at the temperature t', 



, a + b tan ilr 

 tany 1 = — \, 



, , , . x a + Man (i/r + e) 



tan (Yi + e) = IT — , , \ 



KT J c — a tan (1/r + e) 



Eliminating ^ between these two equations and dividing 

 through by tan e, we obtain a linear relation connecting tan yjr 

 and tan e, and thus for any value of yjr we get a real value of e. 

 Thus, considering only the molecular distribution in a common 

 plane of symmetry, the system may be brought from the con- 

 figuration at one temperature to the configuration at any 

 other by (1) a simple linear dilatation along two directions in 

 general not at right angles, or (2) a linear dilatation parallel 

 to any one of an infinity of pairs of lines followed by a rota- 

 tion of the system through some angle round the perpendicular 

 to the symmetry-plane : the lines of one of these pairs are 

 rectangular; and in this case they are the crystal-lines which 

 experience the maximum and minimum elongation. 



