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

 the result 



\ sin r# ar— —r— 1 Err ar sin rx ar — • 



Jo ^Jo * ■ 



in which it is easy to see that we may legitimately put cos oo=0 

 and obtain 



I Erf «r sin r#c?r = -~— ( 1— e ^ 2 \ . . (21) 



This result can also be obtained independently and in a more 

 simple manner by integrating 



Jo 4 " 3 



with regard to a. 



The well-known formula 



I 



e -a 2 ^ cos budget— 



a* + b* 

 affords, on integration, 



1 Erf #v# cos o<2? -7-= I -j — r^ 

 Jo V^ X « 4 + 6 2 



1 f 1, a*-a\/n + b / /2 ,\ 



-tan-i(a^-l); (22) 



and similarly from 



r°° . b 



1 e ~ a2 * sin bxd%= -* — r- 

 Jo a + b 



we deduce 



C™ doe 



I Erf « \/w sin 5a? —7- 



Jo Vx 



= ^mW° S a^a^2b + b +tan (— +1 ) 



^-P?- 1 )} »> 



Many other formulse could, no doubt, be found; but the 

 above probably include the most simple cases. When the con- 

 stants have imaginary values assigned to them, the results sng- 



