438 Mr. R. Pendlebury on some Definite Integrals. 



We get then readily, integrating by parts, 



B "t 



nr 1 f* arc tan- t /» 2 



erf (*) erf (/3) = £ - ± 1 a e - a2sec2 ^0- I I € -^cosec^ # (3) 



4 ^Jo ^Jarctan^ 



Putting in the first integral on the right hand tan 6—oc, and 

 in the second cot0=#, 



B a 



and, in particular, when « = ft 



. lerfW^f-e-^^. ... (5) 



In combination with the last equation may be used the 

 equation (5) of Mr. Glaisher's paper quoted above, which gives 



/*0O -0(2^2 J 



v / ^Erf(a) = e -«j o / TT ^ (6) 



Putting, in (5), erfa=^-ZT-Erf(a), we get 



-v^BrfW + {Erf («,)}•=-«-. J ij^j 



.-. {Erf («)}»=«--)_ ^?. ...... (7)* 



If we multiply both sides of (7) by da, and integrate between 

 the limits < q , we get the curious result, 



* The equation (7) can be obtained directly without much difficulty. 

 For 



i= I e-* 2 cfa?= I 



Erf(«)= ^ 

 „ „: N dErf(«) C«> 



'. e"* 2 Erf («)= J «e-^d+« 8 )^; 



and integrating, 



{Erf(*)f= r^e-^+^daj 



l+# 2 

 the constant of integration being zero. 



