96 On the Dilatation of Crystals on Change of Temperature. 



In exactly the same way it may be proved that, in the case 

 of an anorthic crystal, if the crystal-lines of unit length coin- 

 ciding at a temperature r with fixed rectangular axes in space 

 have at a second temperature r f increased to a, ft, and 7 re- 

 spectively, and have taken new directions denned respectively 

 by the direction -cosines 



\ 1 fi 1 v 1 , XsfWa, \ 3 p 3 v 3 — 



(1) One triad of lines not at right angles will have the 

 same directions in space at both temperatures, each line being 

 determined by the three equations 



(Xi*— 8)x + \ 2 fy + ^7 z =0> 



v x ax+ v 2 (3y + (v 3 y — 8)z = 0, 



where 8 is any one of the roots of the cubic 

 X x a — 8 \ 2 /8 X37 



(XiO. P2/3 — 8 fZf/ =0. 



Via y 3 /3 v 3 y — 8 



In any case one of these lines must be real ; and by the prin- 

 ciple of the " conservation of areas " it may be shown, as above, 

 that in the limit all three must be real. 



(2) An infinity of triads are equiangular at the two tempe- 

 ratures ; one only of these triads is rectangular. This property 

 of two homographic point-systems is virtually proved in a 

 paper by Prof. H. J. S. Smith, " On the Focal Properties of 

 Homographic Figures," published by the London Mathematical 

 Society. The dilatations along the lines which are at right 

 angles at the two temperatures will be maximum, mean, and 

 minimum respectively. 



Thus we may pass from the configuration at the first to that 

 at the second temperature by (1) simple linear dilatations 

 along three lines in general not at right angles, or (2) linear 

 dilatations along equiangular triads followed by a rotation of 

 the system as a rigid body. 



It would appear that the term thermic axes, if applied at 

 all, ought really to be devoted to those which have been here 

 called atropic. 



The following problems may be suggested for solution: — 



(1) What is the relation between these atropic lines and 

 the directions of the edges of an elementary parallelepiped ? 



(2) Are the same lines atropic for all temperatures ? 



