Mr. J. W. L. Glaisher on a Class of Definite Integrals. 427 

 where 



/(8)= , + ( 1 _i) i 4 + ( 1 _i + j)_j : _ + ....; 



Comparing this result with (5), we find 



Erfc.z=e-^V7t^). (49) 



The forms assumed by (5) and (44) when x is taken = tan are 

 worthy of attention from their simplicity ; we have ^ 



e- ai ^ Q dd=\/ire a ^viVa s . . . (50) 



I 



o 

 and 



j\ rf(aS ec0)^ = _^Ei(- ffl 2 ). (51) 

 Writing x 2 for x in (18) , we find the area of the curve y =Erf ax, 



viz. 



1 



f Erf(aff)dfc=~. .... . . (52) 



For the numerical calculation of the values of the error-func- 

 tion, there are three series, viz. 



Erfca;= *-3 + n2Er-i^r3r +•••' .. • • (53) 



J, 2.r 2 (2a: 2 ) 2 (2* 2 ) 3 \ «... 



Erfc*=*e-(l + _ + ^i- 5+r: L_J_ + ...j, (54 ) 



„ , er*/ 1 1.3 1.3.5, \ .,,. 



™ x= iu\ l -w + W)*~-w? + -'-)' • (55) 



and the continued fraction 



***- 2* 1+ 2# 2 + 1+ 2* 2 + TT&cT' ' ' " (5b} 



The formula (34) might also be of use in the calculation of 

 the integral. 



" The discontinuity in the values of the constants and other 

 points connected with the series (55), when x is of the form 

 a + bi, are fully discussed by Professor Stokes in a memoir "On 

 the Discontinuity of Arbitrary Constants which appear in Diver- 

 gent Developments," Camb. Phil. Trans, vol. x. 



Applications of the Error -function to Physics fyc. 



The work which first gave the error-function an importance 

 of the first rank in physics was Kramp's Analyse des Refractions 

 Astronomiques et Terrestres, Strasbourg, 1798. Kramp shows 



