[ 437 ] 



LV. Note on some Definite Integrals. 

 By R. Pendlebury, St. John's Colleye } Cambridge*. 



IN a recent communication to the Philosophical Magazine f, 

 Mr. Glaisher pressed the claims of the integral 



e~* 2 dx (1) 



I 



for admission into the rather too scanty family of known func- 

 tions, and gave a tolerably long list of integrals expressible 

 linearly by this function. The object of this note is to point 

 out that a fresh set of integrals can be found expressible by the 

 squares, cubes, and higher powers of the integral (1). With 

 Mr. Glaisher's notation, 



I write also 



so that 



f 



j; 



6-**^= Erf (a?), 



€- x2 dcc= erf (a?) ; 



V-7T 



Now 



Erf(a')+erfW = — • * (2) 



erf (a) erf (/3) = f V* 8 dx f V^ 2 dy 



Jo Jo 



= j \pe-?dpd0, 



the integral being transformed by the equations w = pcosO } 

 y=p&m6. The new double integral divides into two, with 

 different limits; viz. 



£1 



asec#\ a arc tan- I 



IT 



/3cosec#\ a o 



* Communicated by the Author. 



t Philosophical Magazine, October 1871, 



