92 Mr. L. Fletcher on the Dilatation of 



the centre of mass of the group, the moment of momentum of 

 the molecule about the axis is mr 2 co, and for the whole system 

 2mr 2 &). From the above this must be zero ; and thus at any 

 instant the values of a> for some molecules must be positive 

 and for others negative, and there must be at least one inter- 

 mediate position for which co must be zero. From the above 

 equation, or from more simple considerations, it follows that 

 there is a second real line at this instant fixed in space. For 

 these lines Professor Maskelyne (to whom I may here express 

 my hearty thanks for his many valuable suggestions) proposes 

 the convenient term atropic. The change of configuration of 

 the system at this instant may be treated as a simple linear 

 dilatation along these two directions. Expressed in rectangular 

 coordinates, the above condition %mr 2 co = may be written as 



6%m% 2 —( K <z~^yZmxz—<$%mz 2 = §. . . . (2) 



As the atropic lines are in general not at right angles, and 

 yet one pair of crystal-lines rectangular at one temperature 

 has been shown to be also rectangular at a second, it is 

 seen that these latter lines will in general have changed their 

 direction in space. The position of this pair of lines may be 

 found in the following way: — Let r ^ be the polar coordi- 

 nates of any point P, OX being the initial line, and let co be 



.1 dr 

 the angular velocity of OP at any instant, while - j-, the rate 



of elongation of the unit crystal-line in the direction OP, may 

 be denoted by k. We have immediately, by resolution per- 

 pendicularly to the radius vector OP, 



co == [6 cos yjr + y sin yfr] cosyjr — [a cos yjr — cf) sin i/r] sin yjr, 

 or 



2a> = + 0+[0-<£]cos2^-[>- r ]sin2^ (3) 



Similarly for the line perpendicular to OP, and by hypothesis 

 having the same angular velocity, 



2cq = + <J>— [6— </>] cos 2-»|r + [a— y]sin2a/r. 



Equating these values of w, we deduce 



•=*[*>*], W 



tan 2f= 6 ^i (5) 



a — y 



So long as the atropic lines are fixed in space, the ratios of the 

 coefficients of x 2 , xz, and z 2 in equation (1) must be constant, 

 and just so long will yjr remain constant. If, then, the lines 

 given by equation (1) are atropic for more than one instant, 

 the directions of the crystal-lines which are at right angles and 



