(8) 

 (9) 



262 DR JAS. BURGESS ON 



and taking the integral from t = to t = t, we have 



f* ft ft f ft 



2 f 9 / ft ft f t 9 t 11 \ 



That is- -^J ^ — ^_- + __ — + ^-_+etcJ = H. 



(2) Integration by parts shows at once that — 



[t n dte- fl = ,.t n + l € - ''" + r Ip+'dte-* . 



J n+1 n+U 



And putting successively n = 0, n = 2, n = 4, n = 6, etc., we get by repeated substi- 

 tutions — 



fdte-'"- = te-*+2JtZdte-* = e - t - 2 (t + ^+'^jt i dte-'"- = e-' 2 (t+^ + |^) +^jtHte^\ etc., 



which vanishes when t = 0, and when £ = t, we have — 



(3) By a process similar to the last we find that — 



jt- n dU-''= -U- n - l e~^ -\(n+l)\t- n --dte- t - , etc.. , 



Hence fdte-- = C-'"Ul-±+ M. - L3 ^+ etc\ 



J afe ^ 2^V 2* 2 + (2* 2 ) 2 (2* 2 ) 3 ^ / 



Putting £ = t, the constant quantity is eliminated by making the integral vanish, and 

 we have — 



C T u ., e-'V. 1 1.3 1.3.5 , \ e-* 2 /, 1 , 1.3 , \ 



J t dte = ^t{ 1 -w + wf"wf + etc J- 2t( 1 -2^ + (2^) 2 - etc r 



Then putting t = co , we have the series — 



and r^-.= =i7 T - e ^(i_i_ + J^ 2 _iL5 +etc y = JV7rH (12) 



•/ o 



The series (8) and (10) are convergent, but when t exceeds 2, the convergence becomes 

 very slow. The first (8) and third (11) are alternately greater and less than the integral, 

 so that if we add to any number of their terms the half of the following term, the error 



* Conf. Hymers' Integ. Gale, pp. 123, 151. 



