270 



Mr Glaisher, On a symbolic theorem [May 19, 



and 



+ e 



r°^ o o 63 r „ „ 62-1 °° 



Jo L Jo 



-"'"'-^ (2x'e~^ dx = [ "(6^' + 46^) e-«''^''-S dx, 



therefore 



whence 



dci 



\ = ["(4^2 + m g-'^'^^'-S dx, 

 a Jo 



^,-46^]ii = 4 a^^e-'^-^'-^^c^^ ...(3). 



If instead of the integral u we start with the integral 



U =■ \ X e a;2 dx 



we have 



and integrating the second term by parts as before, we find 



Wft Jo 



so that 



{^-m M, = 4 (71+ 1) j ^''*+'e-«'^^"^-2cZ^ 



= 4(n + l)zf„^j 



Thus from (3) and (4) 



and, writing for ^i its value — e"^"*, this is 



(4). 





(ia/ (X ' 



which on writing x in place of a, and a in place of — 26 is the 

 theorem (1). 



