170 REPORT— 1873. 



Denmark 7 Portugal 1 



Italy 3 Sweden 1 



United States 3 Eussia 1 



Switzerland 2 Egypt 1 



Spain 1 



Belgium supplying none. These figures afford no comparison between Great 

 Britain and other countries ; but they give a fair idea of the relative table- 

 publication of foreign countries, or, at all events, of the relative proportions in 

 which their tabular works are to be found in English libraries. The numbers 

 of tables published in some of the chief towns are as follows : — London 94, 

 Paris 23, Berlin 18, Leipzig 17, Edinburgh 11, Vienna 5, Copenhagen 4, 

 New York 3. Of the 109 works published in Great Britain and Ireland the 

 following is the distribution : — England 96 (London 94, Boston 1, Ci- 

 rencester 1), Scotland 12 (Edinburgh 11, Glasgow 1), Ireland 1 (Dublin), 

 showing the paramount position of London in the publishing trade in this 

 country. 



Art. 13. Contents of the Eeport that was intended to be presented to 

 THE Bb.u)ford Meeting, 1873. — Owing to the great amount of space already 

 occupied in the present volume by the foregoing Eeport, it seemed desirable 

 to postpone for a year the Eeport which it was till recently intended should 

 be presented to the Bradford Meeting, and only to give here a brief 

 description of the work performed in 1872-1873. This latter Eeport (which 

 is not lengthy) consists of three parts — (1) Tables of the Legendrian Func- 

 tions ; (2) List of errors in Vlacq's ' Arithmetica Logarithmica,' 1628 or 

 1631 ; (3) Account of the tabulation of the Elliptic Eunctions. 



Art. 14. The Tables of the Legendrian Functions (La^tlace's Coefficients). — 

 These give P"(a') to n=7 from x=0 to x=l at intervals of -Ol, viz. the 

 functions arc : — 



P' = l, 

 P'=.i', 



P= = i(3.j--1), 



P3 = l(5.r=-3.r), 



P' = l(35.v'-.30.r' + 3), 



P= = ^(63,t''-70.f^ + lo,r), 



P« = JL(231.r'' - 3U.v' + 1 05.v= - 5), 



p'' = yV(^-9'^'' - '^^^•^'' + '^i^-^'' - 2^-^') ; 



and as only powers of 2 appear in the denominators, all the decimals ter- 

 minate, and their accurate values are therefore given. The work was per- 

 formed in duplicate — one calculation having been made by Mr. W. Barrett 

 Davis, and the other under the direction of the reporter, by whom the two 

 were compared, the errors corrected, and the whole differenced. As the 

 accurate values of the functions were tabulated, the verification by differ- 

 ences was absolute. A short introduction on the use of the tables in inter- 

 polation was written by Prof. Cayley, who has also made drawings of the 

 curves y = Y"{x) over the portion calculated. 



Art. 15. The List of Errors in Vlacq's 'Arithmetica Logarithmica' (1628 

 or 1631). — It seemed very desirable that a complete list of the errata in 

 Vlacq, 1628 or 1631, should be foi-med for the convenience of those who 

 have occasion to employ ten -figure logarithms. No less than five copies of 

 this work have been continually in use in the calculation of the Elliptic 



