224 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



numbers in the various difference columns are zero. Now in such functions 

 as ordinarily occur in practice the numerical values of the differences, if the 

 intervals are not too great, decrease with rapidity and run smoothly. If an 

 error is present, however, the differences of higher order become very irregular. 

 10.25 As an illustration, consider the function y = sin t for t equal to 10, 

 15, The following table gives the function and its successive differ- 

 ences, expressed in terms of units of the fourth decimal: 1 



TABLE II 



Suppose, however, that an error of two units had been made in determining 

 the sine of 45 and that 7073 had been taken in place of 7071. Then the part 

 of the table adjacent to this number would have been the following: 



TABLE III 



The irregularity in the numbers of the last column shows the existence of an 

 error, and, in fact, indicates its location. In the third differences four numbers 



1 Often it is not necessary to carry along the decimal and zeros to the left of the first 

 significant figure. 



