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_2 ft _ 2 

 IX. — On the Definite Integral J-w I e dt, with Extended Tables of Values. 



By Jas. Burgess, CLE., LL.D. 



(Read July 15, 1895.) 



1. The integral Je~ t2 dt occurs so frequently in various branches of research that, as 



far back as 1783, Laplace suggested that it would be useful to tabulate its values for 

 successive ranges of integration.* It is employed in investigations on the theories of 

 refraction, conduction of heat, of errors of observation, of probabilities, etc. These are 

 familiar to physicists and need not be dwelt upon, t 



The Integral. — Previous Tables. 



2. The important formula or result — 



€-*dt = iJir. ( 1) 



i 



appears to have been discovered about 1730 by Euler, J who expressed it in the form — 



J ( l0g6 x) ' dx =-J 7r >§ 



2 / 1\ ~" ' 



for, putting x = e~'~, we have (log e -J dx= —2e~ t dt. 



3. Since e - fi dt=\ e - fi dt + e -"dt, (2) 



* Histoire de VAcad. Roy. des Sciences, 1783, p. 434 ; conf. Todhunter, Hist, of the Theory of Probabilities, p. 486. 



t Conf. Glaishee, in Phil. May., vol. xlii, (1871), pp. 429-31. 



% Gauss ascribed this integration to Laplace : Oriani (in Zach's Monatliche Corresp. for March 1810, Bd. xxi, 

 S. 280 f.) pointed out Euler's prior claim, but Gauss did not correct his statement, Theoria Motus Corp. Ccel., art. 177, 

 p. 212, and Werke, Bd. vii, Ss. 233, 280, 289 ; Davis's transl. of Theor. Mot, pp. 258, 259. Legendre (Exercices de 

 Calcid Integral (1811), torn, i, p. 301) asserts Euler's discovery, and refers to his paper, "Evolutio formulae integralis 



J r - l dx(\.x)n," in Novi Commentarii Acad. Scient. Imp. Petropol., torn, xvi, (for 1771) p. 111. Conf. ib., p. 101 ; and 

 Comment. Acad. Scient. Petrop., torn, v, (ior 1730-1731) p. 44 ; also Euler's letter to Goldbach of 8th Jan. 1720, in 

 Fuss, Correpond. Math, et Phys., torn, i, p. 13. 



§ This is the form used by Legendre in his "Traite des Integrales Euleriennes" in Fonctions Elliptiques, etc. 

 torn, ii, pp. 365, 517-524. 



VOL. XXXIX. PART IT. (NO. 9). 2 R 



