Crystals on Change of Temperature. 93 



are rotating in the same direction with equal velocities will 

 also be fixed in space for more than one instant. But it must 

 be noticed that, by reason of the angular velocity of such lines, 

 different pairs of crystal-lines will coincide at different instants 

 with these fixed directions in space : in other words, if two 

 lines are permanently atropic and not at right angles, not only 

 will the lines which at any instant are retaining their perpen- 

 dicularity have an angular velocity, but at different instants 

 different crystal-lines will possess the property. We shall now 

 proceed a step further, and show that this retention of the 

 mutual inclination for any instant is possessed by an infinity 

 of pairs of lines. If % be the angle any crystal-line makes 

 with the axis OX, it follows from (3) that, if co be the angular 

 velocity of this line, 



2ft> = + (£ + (0-<£)cos2 % -(«-7)sin2%. . (6) 

 If yfr, yjr + — be the angular coordinates of those lines which 



at this instant are retaining their perpendicularity, we may 

 write from (5), 



6 — </> = asin2-v/r, «— y = acos 2i|r, 



and thus \ • . 0/ N 



2o> = o + <p + a sm 2(y — %). 



Similarly, if co be also the angular velocity of a line whose an- 

 gular coordinate is %', 



2(0 = + cj> + asm2('y{r—x / )- 

 Equating these values of co, we deduce 



(% / -f) + (%-f)=|- 



Thus, at any instant, any two lines will be retaining their 

 mutual inclination if the sum of the angles which they make 

 with one of the lines of that pair which is retaining its per- 



rrr 



pendicularity be equal to w — in other words, if they are equally 



inclined to and on opposite sides of a bisector of the angle 

 between the lines of this latter pair. 



The following relations between the angular velocities and 

 between the rates of elongation of perpendicular lines are in- 

 teresting. 



If co' be the angular velocity of the crystal-line whose an- 



gular coordinate is %+ ^, it follows, as above, that 



2a/ = + <£-(0-0)cos2% + (a-y)sin2%; . . (7) 



