ON THE CALCULATION OP MATHEMATICAL TABLES. 59 



The Calculation of Mathematical Tables. — Report of the Com- 

 mittee, consisting of Professor M. J. M. Hill (Chairman), 

 Professor J. W. Nicholson (Secretary), Dr. J. E. Airey, 

 Mr. T. W. Chaundy, Mr. A. T. Doodson, Professor L. N. 

 G. Filon, Mr. G. Kennedy, Sir George Greenhill, 

 Professors E. W. Hobson, Alfred Lodge, A. E. H. Love, 

 H. M. Macdonald, and G. B. Mathews, Mr. H. 

 G. Savidge, and Professor A. G. Webster. 



Introductory. 



The grant of 35Z. — including 51. returned as the unexpended part of the 

 previous grant — has been utilised completely during the present year, 

 and the Committee is able to put forward several completed Tables 

 for which there has been a considerable demand among physicists, as 

 evidenced by written requests to the Secretary. Some other Tables, not 

 at present complete, are still in hand, and it is proposed during the 

 coming year to devote more attention to the roots of Bessel functions 

 which are needed for the solution of physical problems. The Committee 

 desires to ask for a renewal of the grant of 30/., especially in view of the 

 fact that their expenditure has exceeded the former grant, on account of 

 the simultaneous completion of several different Tables. The Report 

 may be divided into five Parts. In Part I. there are three Tables of sines 

 and cosines of angles expressed in circular measure. The main purpose 

 of such Tables is to facilitate the rapid calculation of transcendental 

 functions from their asymptotic expansions. They have been the subject 

 of special approval by the Association. Tables I. and II. have been under 

 the care of Dr. Airey, and Table III. of Mr. Doodson. 



Part II. deals with the Bessel and Neumann functions whose order and 

 argument are nearly equal. Dr. Airey, to whom they are due, has 

 recently extended the formulfe of Nicholson and Debye relating to these 

 functions, which are now somewhat prominent in physical work. 



In Part III. Mr. Doodson continues his Tables of Bessel functions of 

 half-integral order, and some of their derived functions. These Tables 

 are a continuation of those in the Report for 1914. 



Part IV. continues the work of Mr. Savidge on Tables of the ber and 

 bei functions and their derivates. 



Part V. contains some valuable Tables of the logarithmic Gamma 

 function and its derivate, together with the integral of the function. 

 These have been calculated and kindly offered to the Association by Prof. 

 G. N. Watson. In recording their appreciation, the Committee desires 

 to suggest that Prof. Watson should be added to their number. 



Paet I. 



Sines and Cosines of Angles in Circular Measure. 



The trigonometrical functions, especially the sines and cosines of angles 

 expressed in radians, are of frequent occurrence in the asymptotic expan- 

 sions of transcendental functions. The only tables hitherto published are 



