LAPLACE, 0I7 



that is, by integrating by parts, 



= [xhy<f>] +j |(1 -^) ^-^ (#)j Bi/dx. 



Wliere by [xBi/^] we mean that xhy^ is to be taken between 

 limits. 



Assume </> such that 



and take the limits of integration such that [^ % </>] = ; then 

 our proposed equation is satisfied. 



From (1 — cc) (^ — y- (x(^) = 0, we obtain 



where ^ is a constant. Then xBi/ (f> will vanish when x= and 

 also when cc = 00 . Thus, finally 



/•CO" 



x^e'^dx. 



Now we proceed to put this integral in the form of a series. 

 The maximum value of ic*e~* is easily found to be that which 

 corresponds to a; = 5. Assume, according to Art. 957, 



23* e"* = s'e~^e~^\ 



8. 



and put x—s-\-Q\ thus 



Take the logarithms of both sides ; thus 



Hence by reversion of series we get 



3 9 V2s 



