Mr. J. W. L. Glaisher on a Class of Definite Integrals. 423 



and therefore, by Fourier's theorem, between the limits and a 



of x, 



Erfar + -Erf(a?-fl) + Erf(ar + fl)+.». 





sin h h sin 1- . . . — e « 2 sin 



a 2 a a 



1 _*£ . 4nrx \ 

 — - e a* 2 sin ... X 



2 a J 



ssZyV - —^ e a2 sin— — \--re « 2 sm — + ... > (32) 



on replacing the former trigonometrical series by its sum 

 1/ 2<7nr\ 



We can either deduce from this equation or prove indepen- 

 dently that 



Erfa?-Erf(ff-fl)--Erf(ff + fl)+ ... 



= 7i -/it 7— { e 4a2 sm {_ e «a sm \. ...}-. (33) 



2 v 7r L a 3 a J 



If ^ be negative in (32) or (33), the sign of the constant JVtt 

 must, of course, be changed. 

 Putting x=^& in (33), we find 



Erffl-Erf3a+Erf5a-... 



1 1 / _^1 1 9i± \ 



= A ^ /7r - -J- \f~ l6a3 "" o 6 l6tt2 + ' ' • )> ( 34 ) 



writing a for J#. 



This formula, (34), I have verified numerically to seven decimal 

 places when a=.\. 



In (29) put #=0, and we have 



1 . /it C 7r2 9* 2 1 



e -«2_ e -4a2 + e _ 9 « 2 _ . . . = i - vZT^ e-4^ + e~«-2-+ . . . L,(35) 



Integrate this between the limits x and 0, and we obtain an in- 

 teresting formula connecting the error-function and the expo- 

 nential-integral, viz. 



V Erfc x-\ Erfc 2x + jErfc 3a?- . . . 



_1 : VV 



~2 X + 2 





since 



%2 



ct 2 



