28 REi'OET— 1889. 



First Report of the Committee, consisting of Lord RATLEion (Chair- 

 man), Professor Cayley, Mr. J. W. L. Glaishee, Professor A. 

 Gr. Greenhill, Professor W. M. Hicks, Professor B. Peice, Sir 

 William Thomson, and Professor A. Lodge {Secretary), ap- 

 pointed for the purpose of considering the possibility of cal- 

 culating Tables of certain Mathematical Functions, and, if 

 necessary, of talcing steps to carry out the calculations and to 

 publish the results in an accessible form. 



The tables -which have first come under the consideration of the Com- 

 mittee are those of the Bessel Functions, viz., the solutions of the diffe- 

 rential equations : 



■^'£'+4><'''-'>=° • • • (^)' 



and 



^'?-^ + ^?+(^'+"> = • • • (2)- 



The two solutions of equation (1) are denoted by ZJ^x) and T„(re). 

 No tables appear to have been made of YJx), but there are several tables 

 of J,^*)- The most complete tables of JoC*) and Jj ix), and the only tables 

 of these functions which are published separately, have been published (at 

 Berlin) during the present year by Dr. Meissel, of Kiel, giving the func- 

 tions to 12 decimal places for values of x from to 15'50 at intervals of 

 001. A shoi't table is given by Bessel ('Uber die planetarischen Storun- 

 gen '), and others are to be found in Lommel, ' Uber die Bessel'schcn 

 Functionen,' and Lord Rayleigh, ' Theory of Sound,' vol. i. Lommel's 

 and Lord Rayleigli's tables were originally calculated by Hansen, and 

 published by him, but the notation adopted by Hansen was different from 

 that now used, What is usually called J„(a;) he denoted by J„(^a;). 

 Lommel's tables give Jo(.t) and J,(.-);) to 6 places from a; = to 20-0 at 

 intervals of 01, and a few values of J„(*) for other integral values of n. 



The two solutions of equation (2) the Committee propose to call \,fx) 

 and K„(a;), in accordance with that adopted by Mr. Basset in the second 

 volume of his treatise on Hydromechanics. They have calculated ^,f^x) 

 for integral values of n from to 11, from a- — to 6 at intervals of 0-2. 

 The calculations are to 12 significant figures, except in the case of \^^{x), 

 some of which are given to 12 and some to 11 figures. The last figure is 

 approximate. A series of 7; zeros between the decimal point and the first 

 significant figure is expressed by 0". It is proposed to interpolate to the 

 interval O'Ol in, at any rate, the cases of Jo(a;) and Ii(a-), and to continue 

 them to higher value s of x, with a view to publishing the various series 

 of functions in book form. 



The Committee desire to thank Professor M'Leod for the temporary 

 loan of his ' Edmondson's Calculating Machine,' and Mr. Walter G. 

 Gregory and Miss B. C. Lodge for considerable assistance in the calcu- 

 lations. 



