REPORTS ON THE STATE OF SCIENCE. 
The further Tabulation of Bessel and other Functions.—Report of 
the Committee, consisting of Professor M. J. M. Hizu (Chair- 
man), Dr. J. W. NicHouson (Secretary), Professor ALFRED 
Lopas, Professor L. N. G. Finon, Sir Gkorcr GREENHILL, and 
Mr. J. R. Arrey. 
Part I.—Hlliptic Functions. 
Tue calculations of the Committee have proceeded steadily during the 
year, and the results are given in four sheets of tables for four modular 
angles, 
Sir George Greenhill has prepared the following statement for the 
explanation of the notation, and of the mode of use of the Tables for 
the various applications which may arise :— 
The Notation and Use of the Elliptic Function Table. 
The Elliptic Integral which arises in a physical problem of Dynamics 
or Electro-magnetism requires to be carried out to a numerical result, 
and with as little delay as possible ; but so far Table IX. in Legendre’s 
‘ Fonctions elliptiques’ is the only source available for reducing the 
labour of the calculation. 
This Table IX. of Legendre gives F(), the First Elliptic Integral, 
and E(#) the Second Elliptic Integral, for every degree of 9, and for 
every degree of the modular angle 6 ; these are defined by 
$ ¢ 
— | 4 = 
(l) 2 ad 
0 0 
(2) A¢g=vV(1—«? sin’), x = sin 4, x’ = cos 0. 
Legendre has shown that F(») and E(¢), together with the com- 
plete functions F(}r) and E(}7), denoted by K and H, are sufficient 
for the numerical calculation of the Third Elliptic Integral, when 
complete; as required, for instance, for Q, the conical angle subtended 
by a circular or elliptic disc, which gives the magnetic potential for 
-uniform normal magnetisation, or the apsidal angle of a spinning-top. 
~ But for the general incomplete Elliptic Integral of the Third Kind 
(E. I. IIL.) the Theta and Eta function of Jacobi is required, and these 
are given in the Table by the function 
(3) AG); Be), ~~ Ce); De); 
defined by 
(4) Diy CEE) A(t) an HB) srs, 00.A, 
@(0K)’ H(K)’ 
(5) C(r)= D90—'7), Bir) = A(90 — 1). 
