44 REPORTS ON THE STATE OF SCIENCE, — 1919. 



A connexion is made with the original Table IX of Legendre, for 

 F<t> in terms of f, by the two outside columns ; of <t> and if/, given in degrees, 

 where 



^=am/K, F</,=/K = JqK, 



90 f 



^=am (1 -/)K, F^={1 -/)K=-gQ-K, 



proceeding by equal increments of r and/, whereas Legendre's Table IX 

 takes equal steps in <j). 



The basis on which these three tables have been calculated is the 

 value of e-'^ which Gauss has given to 51 places of decimals (Werke, 



vol. iii., p. 426). This is the value of q for the modulus — , when 



K=K' and ^=45°. The various powers of q, integral and fractional, 

 required for the q series were calculated to 20 places by the aid of an 

 arithmometer, kindly lent for the purpose by Dr. Western, and are 

 collected in the accompanying table. 



In the case of the lemniscate functions (K=K') all the entries were 

 computed by means of the q formulae, 



®2i=l— 2?cos2a; + 2g*cos4a;— 25°cos6a;+ . . . , 



B.ti=2qi sin x - 2q" sin 3a; + 2g V sin 5a; - . . . , 



Z^i=^2, [sin 2nx{q'' + q"'+q"'+ . • •)]. 

 TT-^ sin nr° 



where u=fK, x=-^irf, sin wa;=sin nr°, cos 2Ma;=cos 2m-°. 



The tables for K=2K' and K=4K' have been determined by trans- 

 formation from the Lemniscate Functions, according to the formula given 

 by Sir George Greenhill in Report iii. 1913 of the British Association 

 Committee on Mathematical Tables. 



The q formulae for the higher values of the modulus, especially that 

 for Zu, are very slowly convergent. From 35 to 40 terms in the series 

 for Zm would be required for each entry in the table for K = 4K' 

 to ensure an accuracy of ten significant figures. Check values for 

 r = 15, 30, 45, 60, 75 have, however, been obtained by the q formulae, 

 and all the tables have been submitted to scrutiny by the method of 

 differences to the fifth and sixth orders. 



The values of K have been obtained from the formula 



K=£, ®o^ 



and E' from 



*^-2L ^^1 V 2.4.6...2r ) 2r-lJ 



but E, for which the above series is slowly convergent, has been 

 calculated from 



EK'+E'K-KK'=i7r. 



