Mr. J. W. L. Glaisher on a Class of Definite Integrals. 299 

 Among integrals of less interest may be noticed 



j (cos/w?- sinpx) log (l + | 2 ) ~- 



and 



\/2p 



^{v*r-2Erf^}, 



(16) 



r^-' ?2(tan ' r+cot Vtan^^ = v / 27rErfyV2, . (17) 

 Jo 



obtained from results given in De Haan's Nouvelles Tables, 



No. 11, Table 178, and No. 9, Table 276. 



There are several simple formulse involving the function under 



the integral sign. Thus from 



we deduce 



from 



C e~ a * . 1 

 Jo ^ a « f 



1 



1 



\/ a 



(18) 



2a 



by integrating between limits, we find 



and from 



we deduce 



I (Erf^-Erf^)-=-^-log-; 



(19) 





2c 



l e -^ 2 Erf-^=^e- 2 -. 



(20) 



then 





By the aid of Fourier's theorem, that if 



/2 C °° 

 y(r) = a / - I <j) (x) sin r# efo?, 



<£ (a?) = a / - I f(r) sin r# dr, 

 we derive from (12), which may be written 



f2r^e~^ 



* / 



V 7T 



r ! sin rx dx= a/ - — s/2 Erf ar, 



