viii INTRODUCTION. 



though in some cases the series expressed in terms of the order preceding or fol- 

 lowing vary less rapidly relatively to a. 



In constructing the following Tables, each term was computed for every .03 of a 

 between the necessary limits, and the accuracy of the computations for all the 

 larger terms was tested by differences. The series were then summed to as many 

 terms as were necessary to give the last figure in the tabular logarithms accurate, 

 and the numerical values of the function checked by differences. Logarithms of 

 these values were then taken and checked in the same way. Next, these logarithms 

 were interpolated to every .01 of a, and those functions of which but few values are 

 ever needed are printed in this form ; the remaining ones interpolated to every .005, 

 are practically without third differences, and in this form they are printed. Nearly 

 all the functions given to every .005 of a are accompanied by the logarithms of 

 their variations for every .001 of a. These variations correspond to the value of 

 the argument opposite to which they are written, and their differences are so 

 small that the second differences of the functions may be taken into account with- 

 out any difficulty. 



Since f (a) is usually a small fraction, f(a) will always be found with a high 

 degree of accuracy. 



16. In printing these Tables, the proof was read by copy, and then the page was 

 stereotyped. Plate-proofs were then tested by taking a new set of differences, and 

 all the errors thus found were corrected in the plates. Proofs from the corrected 

 plates were used in preparing the Supplement. As fast as the pages of the Sup- 

 plement were stereotyped, they were tested by a duplicate computation entirely 

 independent of the first. The series printed on pages 42, 43, 44, were also com- 

 puted in duplicate. 



In conclusion, I must express my obligations to my friend and colleague, Mr. 

 Isaac Bradford, for the valuable aid he has rendered me in preparing the Tables 

 for the press, and in computing a duplicate of the Supplement. 



