Mr, J. W. L. Glaisher on a Class of Definite Integrals. 297 

 also Erf 0= J vAr, so that 



fV* 2 ^=!v'7r-Erfa?, 



We know that 



J/ ^"2^ 

 whence 



2 Vc> 



whence, integrating with regard to c, 



r*> e -c(^+ a 2) ^ w 



I — g dx — Yxias/c 



Jo X + a a 



from (3), and therefore 



n GO ^#2 / 



j ^*-^W.^ ... (5) 



This result is not new ; it is obtained, though in a different 

 manner, in De Morgans f Diff. and Int. Calc.' p. 676. 

 From it we can deduce, by Boole's theorem 



I <f>lx )dx = I <p(x)dx, 



a being positive (Phil. Trans. 1857, p. 780), that 



*lt* of s =— ^^A • (6) 



or, taking #=-, changing the values of the constants, and writing 



z 



x for z } 



~ -±L+~- wiv* . (7) 



j; 



a? 4 + (« 2 -2a)^ 2 + a 2 fla 



o 



Putting a = ^-, we have, as particular cases, 



I 43-2— dx =\/ 7T Erf ^2«c 



dx= /> / — ms/2*c. . . (9) 



(8) 



i 



**+■«' 



