434 Mr. J. AY. L. Glaisher on a Class of Definite Integrals, 



{Encyc. MetropoL "Theory of Probabilities/' p. 451). Pe 

 Morgan remarks that he does not know whether this Table was 

 calculated independently or depends on Kramp's Table ; but on 

 p. 269 of the Jahrbuch Encke says that Table I. was deduced 

 from Bessel's Table of ^e~ x2 dx in the Fundamenta. Bessel, as 

 we have seen, tabulated log 10 (e* 2 -Erf a?); and if Encke's Table 

 was derived from Bessel's, it must have been by interpolation 

 from his Table II. It is more probable, however, that it was 

 derived from Kramp's Table I., from which it can be deduced 

 at once. Table II., Eucke states, was formed by interpolation^ 

 and is probably founded on Table I. Both these Tables, as well 

 as Kramp's Tables L and II., are reprinted at the end of De 

 Morgan's article, previously referred to, in the Encyclopaedia 

 Metropolitana. 



The Table accompanying the present paper gives Erf x from 

 a?=3*00 to a?=3'50 to eleven places, from #=3"50 to # = 4*00 

 to thirteen places, from x =4*00 to # = 4*50 to fourteen places, 

 subject to certain qualifications with regard to the accuracy of 

 the last figure, which will be stated further on. The values 

 were calculated by means of the same difference-formula, viz. 

 (60), that Kramp used. ^Separate Tables of log 10 e~ x2 (— —x^fij 

 fj, being the modulus) and of 



Iog I0 {A-A«*+ ^f^ *»-*^£*} 



were formed. The second of these Tables was differenced 

 throughout, and the gradual change of the differences from 

 •0000430 to -0000427 afforded a very good test of its accuracy. 

 A Table of Erf (x + h) — Erf# was then deduced from the two 

 subsidiary Tables, and was differenced throughout as far as A 3 ; 

 and the regularity of these last differences proved the correctness 

 of the Table. Erf 3 was calculated, as previously mentioned, 

 from the continued fraction and the differences subtracted from 

 it, till Erf 3-5 was obtained =-000 000 658 5487... The correct 

 value obtained from the continued fraction was 



Erf 3-5 = -000 000 658 553 76.. ., 



so that eleven figures are the same in both ; it is probable, there- 

 fore, that the last figure in the values from x = 3'00 to a? = 3*50 

 is nowhere in error by so much as a unit. As an additional 

 verification, Erf 3'2 was calculated from the continued fraction 

 and found = -000 005 340 191 ... , which agrees with the num- 

 ber given in the Table. Erf 4 was calculated from the continued 



