EXPERIMENTS ON KAOLINIZATIOK 305 



proximate form, and calculating the constants by the method of least squares. 

 In the present case there is as much reason to adopt a linear form of func- 

 tion as any other, and this would have the advantage of greater simplicity. 

 Denoting the number of hours which have elapsed since the beginning of 

 the experiment by w, let 



Jt=a + f:lu (1) 



In this equation the constant a is without great interest. It simply 

 denotes the value of Jt Avhen u is zero, but is largely influenced by the 

 normal difference of temperature between the interior and exterior of the 

 rock-chamber, i. e., the difference which may be recognized by an inspection 

 of the foregoing tables, and of Table XIV. /?, however, is of importance, 

 representing the increment of temperature of the rock per hour in conse- 

 quence of the T. E. K. It will be noticed that /3 is either negative, positive, 

 or zero, according as the process of kaolinization produces or absorbs heat 

 perceptibly, or is without appreciable thermal effect. In making the calcu- 

 lation for /? I had hoped to be able to derive this constant from the four 

 weeks' observations, as a whole. The problem is difficult, however, inso- 

 much as the results obtained do not form one continuous series. The problem 

 is not, in other words, that of a single straight line as in equation (1), but 

 one involving three straight lines, for all of which, however, the value of 

 /3 is the same. Expressing the whole interval during which the observations 

 were made (four weeks) by 7r\ and regarding the values of M in Table III. 

 as being ordinates of the component line whose extreme abscissae are 

 and J , those in Table VI. as belonging to the line between I and I, and those 

 in Table VIII. to the line between ^ and tt; then the whole line between 

 and /T, expressed as a special case of Fourier's series, would be represented 

 by the equation 



z/i=:^jsin it + ^osin 2m+ 4-^,»sinw»M + - • . 



where 



A^-= - < / ■* («! -f /? ^) sin »i (pdg)-\- / - (a.,-^ /3 cp) sin m cpdip 

 + /. (^3 + A ^) sin m (pd (p > , 



'The assumption furnishes the constant for the reductiou of the observations. 



