EXPLANATION. V 



It is especially convenient when the fourth digit is unity, in which case the 

 product from the tables is added to the three-digit factor to obtain the required 

 product. 



Example: 0.302 times 1227. At first neglect the 1 in 1227. Under .302 in 

 the top line find 227 or the nearest number to 227 and smaller, in this case 227, 

 and opposite 227 is 69. Then .302 X 1227 = 69 + 302 = 371. 



INTERPOLATION. 



Mathematical tables give values of the function for certain values of the 

 variable argument. Values of this function corresponding to intermediate values 

 of the argument are obtained by the process known as interpolation. The con- 

 secutive given values of the argument usually differ by a constant quantity; if 

 this difference be regarded as unity, an intermediate value of the argument will 

 differ from the given value on either side by a fraction. In the process of inter- 

 polation it is necessary to obtain the product of this fraction by the proper varia- 

 tion of the function. By means of the present tables this product may be obtained 

 in a most rapid and accurate manner when the numbers involved are of three digits 

 or less. The variation to be used may be given directly or it can be derived from 

 the given values of the function in the manner explained below. 



Let the quantities be denoted by the following scheme : 



where 



a n denotes the argument from which the interpolation is made ; 



a_i, the preceding argument to a ; 



!, the following argument to ; 



/(a ), the value of the function corresponding to ; 



f(a _0, the value of the function corresponding to a_i; 



/(i), the value of the function corresponding to o^; 



Let 



A" - A' A ' 



- ^1/2 ~ ^-1/2 



Aiii Aii Aii 



A- 1/2 =- A - A_i, 



Aiv . Aiii A'" 



&Q ~ til/2 ~ ^-1/ 



Aiii . 



Ai/2 



Aii 



- A 



, 



a denote the value of the argument for which, f(a), the value of the function, 



is to be derived by interpolation; 

 h = a Q a_i = i a , etc., the common difference of the argument; 



n = 



a ao 



h 



, the interpolation factor 



v, the proper variation obtained from the differences by some one of the formulae 

 given below, the formula applicable in each example being determined by 

 the numerical size of n and of the various differences. 



