PREFACE 



These tables were inspired by the similar work of Sir George Greenhill 

 and Col. R. L. Hippisley as published in Smithsonian Mathematical Formulae 

 and Tables of Elliptic Functions. The pattern they designed has been followed 

 closely by us. The chief difference is our inclusion of the three elliptic func- 

 tions, sn(u,k), cn(u,k), dn(u,k). 



Columns A and D were computed first. From them were computed columns 

 sn(u) and cn(u), checked by means of 



Check I: sn^(u) +cn-(u) =1, with a maximum error of ±2 in the 15th 

 decimal. Practically all errors in these first four columns were eliminated at 

 this time, but some few managed to elude us. 



Column dn(u) was then computed, and checked by means of 



Check II: sn(K— u). dn(u) =:cn(u), with a maximum error of ±2 in 

 the 15th decimal. Those few errors that had eluded us in Check I were dis- 

 covered at this time. 



Column E(u,k) followed, computed by the formula 



E(u + v)=E(u)+E(v)— k2sn(u+v)sn(u)sn(v), 



where u was taken as r/90 K and tabulated r°, while v was taken as 1/90 K, 

 i.e. 1°. Independent computations for r° = 15°, 30°, 45°, 60°, 75°, 90° were 

 used as Check III, with a maximum divergence of ±3 in the 15th decimal. 

 In this method of computing the E(u,k) column the errors are additive, and 

 hence Check III is more sensitive than Checks I and II. 



The ^ column was computed last by means of sin ^ = sn(u), using Andoyer's 

 15-place Table of Natural Sines and Cosines. This was checked by computing 

 cos <^=:cn(u). Interpolation into Andoyer's Tables was not used. Instead 

 we developed a rapid method, fitted to machine computation, based on the 

 two formulas 



sin(<^+c) =:sin ^4-«cos<^ sin ^ cos <^ 



2 6 



cos(^4-£) =cos <^ — € sin (f> cos <f>-\ sin <^ 



2 6 



Maximum divergence here was kept to ±4 in the 15th decimal. 



Our manuscript columns are all, thus, correct to ±4 in the 15th decimal. 

 In printing, these have been cut to 12 decimal figures, assuring the accuracy 

 of the 12th decimal digit. 



Galley proofs were checked by adding entries in each column of our manu- 

 script in groups of ten and subtracting the entries of the corresponding groups 

 in the galley proofs and demanding that the remainder be equal to zero. 



To teachers of mathematics and amateur computers, elliptic functions is a 

 pleasant field of endeavor. Our present tables are not exhaustive. Three more 



