490 WRIGHT: CHANGE IN ANGLES OF QUARTZ 



finally regains its original sharpness but has usually shifted its 

 position noticeably in the field. 



The form of the curve below 575° (a-quartz) suggests an expo- 

 nential curve. To test this conclusion the first, second and third 

 derivatives were formed by the method of differences between 

 values at constant intervals on the curve. The curves of these 

 derivatives were similar in shape to the original curve, thus prov- 

 ing that the original curve can not be adequately represented by 

 an ordinary polynomial equation up to the fifth degree. Accord- 

 ingly an exponential function of the form 



y = at + b (e* - 1) 

 was tried with the following results, the constants a and b having 

 been ascertained from the observed values by the least square 

 method :^ 



2/=51°47'.4- 10.0113173^ + 0.01335(6'^^""- l)i 



Expressed in absolute temperatures this equation becomes 



?/ = 51°51'.4-|o.0113173r + 0.00087093e^^°| 



* This function is the simplest combination of an algebraic with a transcenden- 

 tal function. The introduction of the exponential series into the equation has the 

 effect of rapidly increasing the slope of the curve near the inversion point. On 

 forming the second and third differential quotients of this function we find that 



—— • = -Tz . Now in mechanics the first differential quotient is termed the veloc- 

 ity and the second differential quotient, the acceleration. The product, mass times 

 acceleration, defines the force acting. If we interpret the above differential quo- 

 tients from this view-point we find that the rate of change in acceleration of 

 angular velocity with temperature at any time is equal to the acceleration (force) 

 itself. This condition implies a force which increases with greatly increasing 

 rapidity as the temperature rises and indicates that at some definite temperature 

 (575° in quartz) the force has reached such a magnitude that the crystal forces can 

 no longer withstand it; a profound change in the internal arrangement results to 

 relieve the stresses set up. Simple inspection of the curve indicates that its rise 

 near the inversion point is so great that such a condition can not continue far 

 above 575°. 



It is interesting to note that the exponential function above is similar in form 

 to that suggested by Dr. Adams in the last number of this Journal, p. 469. All 



d^y d-y 



such equations reduce on differentiation to the form -y-^ = K t^ and indi- 

 cate clearly that the curve they represent can not be expressed by a simple poly- 

 nomial with positive exponents. The introduction of the exponential function 

 provides more effectively for the extra rapid rise or fall and consequent straighten- 

 ing out of the curve than the addition of an extra term to a polynomial series. 



