Thermo-electric Interpolation Formula. 469 



„_ log(So"e + /3)-1og(Sge + /3 ). 

 log t" — logr' 



(9) 



m 



W~ or ~w~ (10) 



By means of these the numerical values of the constants 

 may be calculated from those of r f , t", 2o e, &-., as follows : — 



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

 unless some better approximation is in some way suggested. 

 Substituting this value in (8), compute the corresponding 

 value of /3. 



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

 and compute 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 ft. 



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 

 approximations have been made an inspection of the rate will 

 enable the computer to estimate values of n which will be 

 nearer than the preceding approximation, and thus hasten the 

 computation. 



Where an equation is to be computed to best represent a 

 progressive series of observed values of t and % e, this method 

 is of course open to some objections, since it incorporates in 

 the constants the accidental errors of the selected observa- 

 tions from which the constants are deduced. This difficulty 

 can be sufficiently overcome by computing residuals between 

 the equation and the data, and amending the equation if 

 necessary to give them a better distribution. 



Logarithmic Formula. — A very simple expression for inter- 

 polation is of the general form 



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

 short range, t" —t', when ^ — 0° is not less than one third of 



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) 



Iog2o£=wlog£ + logm. 



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



