432 Mr. J. W. L. Glaisher on a Class of Definite Integrals. 

 means of the formula 



3 6 



^-l^ + 3 i4 _ J ■ m 



+ 



30 



obtained at once by Taylor's theorem ; h was = '01, and the 

 term in h 5 (i. e, the last written above) was small enough to be 

 neglected. Kramp does not state what value he started from in 

 applying the differences, or what means of verification he adopted. 

 In all cases where a Table is constructed by means of differences, 

 the last value should be calculated independently, and then the 

 agreement of the two values would verify all the preceding por- 

 tion of the Table. This, however, Kramp does not appear to 

 have done, though it would not, as Kramp intimates, have been 

 impossible. To calculate Erfc 3 from (53) would have been a 

 heavy, but by no means exceptionally heavy piece of work. The 

 remark that the " void " cannot be filled up by any known me- 

 thod is not now true. Laplace's continued fraction gives with 

 very little trouble Erf 3 = 000 019 577 193 .. . This value of 

 Erf 3 differs in the tenth and eleventh figures from Kramp's 

 result obtained by differences ; so that it is probable that a por- 

 tion of his Table is incorrect in the last two figures. I have not 

 yet examined the cause of the error so as to be able to state 

 whether it is due to an inaccuracy in the calculation of a differ- 

 ence, or to an accumulation of small errors in the differences ; 

 but I hope shortly to be able to make a complete examination of 

 Kramp's Table. 



The next Tables of the functions which were published are 

 due to Bessel; they occupy pages 36, 37 of the Fundamenta 

 Astronomic, Regiomonti, 1818. Bessel remarks that Kramp's 

 Table does not go beyond % = 3, and that in a few cases the 

 function might be wanted for larger arguments ; he therefore 

 tabulates e* 2 Erf# for the argument log ]0 # from to 1 (so that 

 the limits of x are 1 and 10) . Bessel gives two Tables. Table I. 

 contains log 10 (e^Erfa?) from a? = to a?=l, at intervals of *01, 

 to seven figures, with first and second differences, and is the same 

 as Kramp's*. Table II. gives log 10 (e* 2 Erf oc), corresponding to 

 the argument log 10 #, the arguments increasing from to 1 at 

 intervals of "01. First and second differences are added. It is 

 not stated how this Table was calculated ; the values correspond- 



* Bessel probably recomputed the Table, as several of his values differ 

 from Kramp's in the last figure. 



