REPORTS ON THE STATE OF SCIENCE. 



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

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

 Professor J. W. Nicttolson (Secretary), Dr. J. K. Airey, 

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

 G. FiLON, Sir George Greenhill, Professors E. W. Hobson, 

 Alfred Lodge, A. E. H. Love, a)id H. M. Macdonald, 

 Mr. H. G. Savtdge, and Professor A. G. Webster. 



Op the grant of 301. given to the Committee, 251. has been expended on 

 the special purposes for which it was allocated — further calculations in 

 connection with the I, Y, and K Bessel functions, the particular calcula- 

 tions selected being in accordance with the frequency of requests for 

 these Tables received by the Secretary from workers in physical science 

 and engineering. The order of calculation is being arranged in accord- 

 ance with the real urgency of the Tables, and the stage is now coming 

 in sight at which the Committee will bo able, as authorised already by 

 the Association at the appropriate time,, to publish, under the auspices 

 of the Association, a volume of fairly complete Tables of the more im- 

 portant transcendental functions. 



The remaining 51. of the grant has been returned to the Association, 

 but the Committee desires to make application for it again, and also 

 for the continuance of the grant of SOI. for the ensuing year. The un- 

 usual circumstances of the past year temporarily hindered the work, 

 so that the Tables for which this particular part of the grant was estimated 

 are still incomplete, but they are well in hand, and some expense has 

 already been incurred in connection with them. 



The present Report contains some Tables of which Dr. Airey has 

 been in charge, divided into three sections : (1) The Bessel functions 

 J„{x) for various orders and arguments which will be apparent from the 

 Tables themselves, their most valuable portions being the entries for 

 which order and argument are approximately the same. These Tables 

 indicate the characteristic tendency towards zero when the order exceeds 

 the argument, as in the formulae of Nicholson and Debye, which, how- 

 ever, were not employed in the numerical work. 



(2) The Neumann functions of types G and Y to a large number of 

 significant figures. These are sufiicient to form the basis of a rapid calcula- 

 tion of complete sets of Tables for functions of these types of any order. 



(3) Continues the calculation of the functions of type Y from (2) as 

 a basis. The Committee is greatly indebted to Dr. Airey for taking 

 charge of this important work. 



The functions ber x, bei x introduced by Lord Kelvin, together with 

 their derivates, are of great importance in connection with alternating 

 currents. In the Report for 1912, the Committee pubbshed Tables of 

 these functions and their derivates, calculated by Professor Webster. 

 They are now able, in Table IV. of the present Report, to give the neces- 

 sary supplement to these Tables — the real and imaginary parts of Ko(x i*), 

 with their derivates — the work having been kindly undertaken and 

 carried through by Mr. H. G. Savidge. For a very complete and general 

 account of the mathematical properties and uses of these functions, 

 reference may be made to a paper by Dr. Alexander Russell (' Phil. Mag.,' 



