258 DR JAS. BURGESS ON 



the integral may be taken as separated into two parts — 



(1) J e~ r 'dt, which Mr J. W. L. Glaisher calls the Error-function complement, and 

 indicates by ' Erfc' And — 



/■oo 



(2) I e-r-dt, which Mr Glaisher proposes to call the Error-function, denoting it 



by ' Erf.'* Mr R. Pendlebury accepts Mr Glaisher's name for the second, and writes 

 the first as ' erf,' — which might lead to mistakes. t For convenience of reference, we 

 ma}^ indicate the second by G. 



And we shall put for the multiple of the first function here dealt with — 



H = -2-|e-'VB. (3) 



Whence, from (2)— H = l-_ ^-G; and G = ^J£ (1-H). (4) 



N /7T 2 



And since hjir= 0-886 226 925 452 758 013 649 083 741 670 , 



and its reciprocal— -L = 1-128 379 167 095 512 573 896 158 903 120 , 

 also log * -=0-052 455 059 316 914 268 038104 750 579, 



"V7T 



it is comparatively easy to derive the value of G from that of H, or the converse. 



4. In 1789, M. Kramp, in his Analyse des Refractions, was the first to tabulate G 

 from Z = 0*00 to t = 3'00, for every hundredth of a unit, together with the logarithmic 

 values and differences. To these he added a third table of the logarithmic values of 

 e< G = e' I e~'~dt, which is useful in connection with the theory of refraction. Kjkamp 



apparently computed the earlier part of his table by the usual formula (8) given below; 

 but it converges so slowly for values of £>1, that Kramp employed a difference formula 

 — to be referred to later — in order to fill up and complete his table. For the lower 

 values of t his results are carried to eight places, and are generally quite accurate ; 

 from t = 2 to t = 3 the values are carried to eleven places, and for the last he gives 

 G= -00001957729 in the table and '00001957669 in the text,J— the true value being 

 •00001957719 3236779. 



Bessel, in discussing the theory of refraction in his Fundamenta Astronomise (1818), 



/■oo 



pp. 36, 37, next gave two tables,§ the first of log. e'j e~ l "dt from t = to ^ = 100, 



J t 



* Philos. Mag., vol. xlii., 4th ser. (1871), pp. 296, 297, 421. 



t Ibid., p. 437. If either is to be called " Error-function," it would seem to apply rather to H than to G. 

 X Twice, pp. 134, 135. 



§ In March 18KJ appeared Gauss' Bestimmung der Genauigkeit der Ikobachtungen, in which he employs several 

 of the constants dependent on values of II. — WerJce, Bd. iv, Ss. 110, 111,1 10. 



