THE VALUES OF -%- (' e - '* dt . 259 



agreeing in the main with Kramp's third table, but differing occasionally in the last, or 

 7th, figure. This may have been due to some recomputation in places where the third 

 differences were irregular. His second table is a continuation of the first, employing as 

 arguments log w x, from to 1 at intervals of '01, with first and second differences. This 

 is equivalent to a short table of log 10 (e ( G) from t = 1 to t = 10, arranged at intervals in a 

 geometrical proportion of which the ratio is — 



t x log-'Ol = t x 1-023 292 992 281. 



It is not explained how this table was computed. 



The next table of the kind appeared in Legendre's " Integrales Euleriennes" (1826), # 

 giving 130 values of 2G, computed to ten decimal places, and arranged in two parts. 

 The first contains the values from t = 0'00 to £ = 0'50, computed by the usual series, 

 and by halving the values of the integrals we can readily verify or correct the 

 early part of Kramp's first Table. The second part is adapted to Euler's form of 

 the integral, viz. — 



j(log.I) 'dx: «=(log.i)\ 



and is arranged with x as argument, from a? = 0'80 (that is, t = 0'472 380 727 077) 

 to ic = 0'00 or t=oo. But, though when x = 0, t is infinite, — in the previous entry, 

 x = 0'01 makes t= Vlog 6 100 = 2*145 966 026 289, — so that this table does not really 

 cover the extent of Kramp's. It was computed by quadratures, and the process is labor- 

 ious and effected by means of logarithmic tables extended to twelve decimal places.t 



In his "Theory of Probabilities"! (1837), De Morgan reproduced Kramp's table of 

 this integral (G) without revision. Mr Glaisher, in the Philosophical Magazine for 

 December 1871, § has further extended it from £ = 3'0 to £ = 4'5 at intervals of 0'01, to 

 eleven places for the first fifty values, thirteen for the next, and fourteen for the last fifty. 

 It would be easy enough to compute it in the way indicated below for any higher values 

 of the argument (§ 25). 



5. But it is with the other integral that this paper is concerned, viz. — 



2 r , 2 r . 



^J J") t 



A table of this was first published by Encke, in a paper on the Method of Least 

 Squares, in the Berliner Astronomisches Jahrbuch for 1834, || giving the values of the 

 integral, for the arguments t = to t = 2 '00 at intervals of 0'01, computed to seven deci- 

 mal places, with first and second differences. This table, the author says, was derived 



* In his Traite'des Fonctions Elliptiques et dcs Integrales Euleriennes, torn, ii, pp. 520, 521. 



t Op. cit., torn, ii, pp. 517-524. The method explained below (§ 12) is different. 



X In Encyclopaedia Metropolitana, vol. ii, pp. 359-458. He also gave a short abstract of it in his Differential and 

 Integral Calculus (1842), p. 657. 



§ Vol. xlii, 4th ser., p. 436. 



I| The paper is continued through the vols, for 1834 (Ss. 249-312), 1835 (253-320), and 1836 (253-308). The 

 Table is in the Jahrbuch for 1834, Ss. 305-308. 



