ON THE CALCULATION OF MATHEMATICAL TABLES. 43 



The Calculation of Mathematical Tablcf;. — Report of the Com- 

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

 Professor .T. W. Nicholson {Secretary'), Dr. J. R. Airey, 

 Mr. T. W. Chaundy, Professor L. N. G. Filon, Sir G. Green- 

 hill, Professor E. W. Hobson, Mr. G. Kennedy, and Professors 

 Alfred Lodge, A. E. H. Love, H. M. Macdonald, G. B. 

 Mathews, G. N. Watson, and A. G. Webster. 



Report on Mathematical Tables of the Elliptic Function. 



Ckiticism was invited of the arrangement of the system proposed of 

 tabulation, and of the accuracy of the tables calculated as specimens, in 

 the Reports 1911, 1912, 1913, on the Elliptic Function. ' 



Some fault was found with the final decimals in the table of the 

 Lemniscate Function K=K', 0=45°, p. riO, Report 1912. 



As this table has been used as the base of the calculation of the table 

 for 



K=(2, 3, 4, 5, 7, . . . .) K', 



by means of a transformation of the order 2, 3, 4, .... as explained, 

 p. 88, Report 1913, it was decided to make a recalculation to a higher 

 accuracy, going to 10 decimals. 



This was undertaken by Colonel R. L. Hippisley, and is given here 

 in Table I, with a description of the formulas of the series employed ; 

 and also of the independent verification by the division values, trisection, 

 quinquisection, and others, given by expressions of finite form. 



The notation proposed in Report 1911 may be recalled here, of the 

 six functions tabulated. A, B, C, D, E, F, with r=90/, 



^^'> 00' ^' HK ' 



C(r) = D(90-r) = ®(l-/IK B(r) = A(90-r) =:H(W)K 



in terms of the theta and eta function of Jacobi. 

 The elliptic functions are then given by 



Jacobi 's zeta function is given by 



zn/K = E(r), zn(l-/)K = F(r) .-: E(90-;). 



The table runs down to 7=45, and then up again to r = 90, as in the 

 ordinary circular function, 



