HOLM AN. — THERMO-ELECTRIC FORMULA. 197 



By means of these the uumerical values of the constants may be 

 calculated from those of /, /', I'o e, etc., as follows : — 



1. Assume as a first approximation some value of n, say n =1, 

 unless some better approximation is in some way suggested. Substi- 

 tuting this value in (8), compute the corresponding value of ^. 



2. Using this as a first approximation, substitute it in (9) and com- 

 pute the corresponding value of n. 



3. Using this value as a second approximation to n, insert it in 

 (8), and compute a second approximation to j3. 



4. With this compute a third approximation to n, and so continue 

 until consistent values of (3 and n are found to the desired number of 

 figures. Then compute m by (10). 



The rate of convergence is not rapid, but after one or two approxi- 

 mations have been made an inspection of the rate will enable the 

 computer to estimate values of n which will be nearer than the pre- 

 ceding approximation, and thus hasten the computation. 



"Where an equation is to be computed to best represent a progres- 

 sive series of observed values of t and 2 e, this method is of course 

 open to some objections, since it incorporates in the constants the acci- 

 dental errors of the selected observations from which the constants are 

 deduced. This difficulty can be sufficiently overcome by computing 

 residuals between the equation and the data, and amending the equa- 

 tion if necessary to give them a better distribution. 



Logarithmic Formula. — A very simple expression for interpola- 

 tion is of the general form 



2o e = m r, 



where m and n are constants. This serves fairly well for a short 

 range, t" — t\ when t' — 0° is not less than one third of «" — t'. 



The convenience of the expression arises from two facts : first, that 

 its two constants are very easily evaluated either by computation or 

 graphically from the logarithmic expression (whence the name) 



log So e = n log t + log m ; 



second, that its logarithmic plot is a straight line, since this expression 

 is the equation to a straight line if we regard log Sq e and log t as the 

 variables. If, therefore, a series of values of 2 e and t are known for 

 a given couple, points obtained by plotting log t as abscissas and log 

 2 e as ordinates should lie along a straight line. Thus a couple may 

 be completely " calibrated " for all temperatures by measuring 2 e and 

 t for any two values of t (suitably disposed). The constants m and n 



