ON MATHEMATICAL TABLES. 171 



Functions (see next article) during the last year ; and it is tlie ten-figure 

 table chiefly used. Besides this, the errata in Vlacq are known with more 

 certainty than are those in Vega, 1794. 



This list had only been partially formed when it was determined to post- 

 pone the Heport ; and it is believed that the year's delay may possibly result 

 in its being made more complete. It is proposed to add a list of errata also 

 in Dodson's ' Antilogarithmic Canon,' 1742 (§ 3, art. 14), and perhaps to 

 consider the subject of errors in tables generally. 



Art. 16. The account of the Tahidation of the Elliptic Functions. — In Sep- 

 tember 1872 it was resolved to undertake the systematic tabulation of the 

 Elliptic Functions (inverse to the Elliptic Integrals), or, more strictly, of 

 the Jacobian Theta Functions which form their numerators and denomi- 

 nators. 



The formulae are : — 



2K.t' 

 e — -= 1 — 2o cos 2a; + 27* cos 4.r—2o^ cos 6.r+. . ., 

 7r 



^ 2Kx 1 ^2K.r 



TT Jc^ IT 



1 / 1 . a 25 \ 



= —A 2q'^ sm 0-'— 25"* sin 3x-\-2q ■» sin 5.r — . . . J, 

 ^* COS ^ + 22-^ cos3.r + 22 ■»cos5.r + . . .), 



=(',)^c%i 



= k'^ (1 + 22 cos 2^+22* cos 4:X-[-2q^ cos 6a? + , . .) ; 



80 that 



2X.V ^ 2Kr . ^2Kx 

 Sin am = Q^ —Q , 



TT TT TT 



cos am = 0., -i- 6 , 



TT ' TT It 



2Xx' „ 2K.i; . ^2K.r 

 A am =63- — ~" J 



TT IT IT 



irK' 



2 being, as always, e ^ ; and the tables, when completed, will give 

 9, Gj, e^, 63 and their logarithms to eight decimals for 



x=r, 2°, . . . 90°, ^- = sin 1°, sin 2°,... sin 90°. 



The tables are thus of double entry, and contain eight tabular results for 

 each of 8100 arguments, viz. 64,800 tabular results. The arrangement will 

 be so that over each page h shall be constant ; and at the top of each page 

 certain constants (i. e. quantities independent of x), such as 



K, K', J, J', E, iK {^f, (^.)^ q, «fec., 



