A \EW CURRENT WEIUHER, ETC. 511 



Putting c'/kf = sin ft, the quantity (F n) can be expressed in terms of complete 

 and incomplete integrals of the 1st and 2nd kinds* ; thus 



c-'*" sincosy8(F-n) = -w-F(*) F(jf, )+(*) F(Jb, ft)+(k)E(V, ft) . (15). 



The various elliptic integrals required in equations (13) and (15) were calculated 

 in three ways, viz. : 



(a) by interpolation from LECKNDKU'H tables ; 



(b) directly by successive quadric transformation ;t 



(c) directly by series. J 



Method (a) was used by two of us independently, one (F. E. ti.) employing 

 a calculating machine, and the other (T. M.) using logs. To obtain the desired 

 accuracy, 1st, 2nd and 3rd differences were required in the interpolations. 



As a check ou possible misprints in the tables, one of us (T. M.) calculated all the 

 complete integrals directly by series, and also both complete and incomplete, by 

 method (i). When the numerical coefficients in the series had been evaluated, the 

 method (c) proved quite expeditious. For the convenience of others who may not 

 have access to tables, these coefficients and their logs are given in Appendix A. 

 Successive quadric transformation, however, proved quickest when the angle ft was 

 well conditioned, three or four transformations being sufficient. But in the case of 

 M e , the angle ft was nearly 45, and to obtain the seventh figure accurately ten-figure 

 logs were used. 



For any particular value of M the corresponding increment coefficients </, r, and .<? 

 are given by the expressions 



(16). 



Denoting -J-? * . , by Z, these may be written 

 - 



where 



* CAYLEY, ' Elliptic Functions,' 183. 



t CAYLEY, Chapter XIII. 



t CAYLEY, Chapter III, 77. 



J. V. JONKS, Roy. Soc. Proc.,' voL 63, pp. 200, 201. 



