Mr. J. W. L. Glaisher on a Class of Definite Integrals. 425 



The Table would be of double entry, and its calculation would 

 entail more labour than the importance of the function at present 

 merits. By the comparison of the two forms for 



Erf (« + W)+Erf (a-fli) 



just referred to, we have incidentally* (putting c=l) the two 

 theorems 



e b2 \ e-* 2 sin2bxdx = e- a2 I e x2 cps2axdx } . . (39) 



Ja Jo 



e b2 f e~ x2 cos 2bx dx + e~ u2 f e* 2 sin 2ax *fo = Erf «. . (40) 



1 b l b 



Denoting for brevity -y-g4«Erf 9 / by u, we see from (10) 



/ d\ n /d\ 2n 

 that f — 1~) u —\~Jf) w, which on taking 2 \/a = u } and replacing 



a by a } assumes the form 



f-jr) e^ Erf- = a [ j- - e<* 2 Erf -, 



\doJ a \ adaJ a a 



or, as we may write it after an obvious transformation, 



^)V* a Erffli==^2fl8^) n flc a?4a Erfa*, . . (41) 



a good instance of a result obtained at once from a definite inte- 

 gral, but which it would not be easy to prove otherwise. 

 Among miscellaneous formulae may be noticed 



f Vi-M {ex) f = - ~ Ei (~2«c) = jT^Erfg) J (42) 

 pErf <M?-Erf te)cos2ra ~= ^{W- ^)-Ei(- ^j\ (43) 



c 



ErfvV + « a ) V(^ + ^ ) = ~ T Ei { -~ a * c) ' ' • (44) 



deduced by integration from tie formula intermediate to (19) 

 and (20), from (11), and from a formula intermediate to (4) and 

 (5) respectively. 



From No. 17, Table 267, and No. 1, Table 266 ofDe Haan's 

 Nouvelles Tables d' Integrates definies, we deduce by dividing by p 

 (p* having been previously written for p in the latter integral), 

 and integrating with respect to p between the limits oo and q, 



* On page 301,, bottom line but one, e~ 2a2c should be e~ a2c . 



