118 Proceedings of the Hoyal Irinh Academy. 



As repeated integration by parts for any term gives 



eK(t-t)jy'^{x)dif = c>^i- 



e-A<-(i)'«-i + XD'"-=^ 



X"-')a; 



J n 



i^xdr 



= e^i 



..A-;.^. 



+ X" 



eHt-nxdt\ 



.11; 



we obtain the general formtila 



leHt-ni,{ir).r.df 



= e^f 



ly -\ 



+ ^(\) 



,eW-f)xdI; 



(12) 



accordingly, if x satisfies (10), so that the left-hand member is zero, the right- 

 hand member of this is also zero for all values of X. 



Let us multiply both sides by dX/6(\) and integrate roimd the same 

 infinite contour. The latter term on the right gives zero. The first expres- 

 sion on the right at the lower limit, t' = 0, has the value 



<PiI>) - 4>W 

 D-X ' 



e\t 



t=o 



(13) 



i.e., the same expression as (4). At the upper limit, f = t, it is 



i>(D) - »( X)^. , 



I) - X "■' t' (14) 



there is one term of order r - 1 in this, viz., ar^'hct, and all others are of 

 lower order ; consequently, when multiplied by d\l<p{\), the integrand tends 

 asymptotically to xtdX/\, and thus the integral is 2Triri:,. Thus we obtain 

 directly the equation 



2iruvt - 



c] 



e'^idX 



0(X) 



X^) - »(X) ,. 



( = 



= 0. 



(15) 



§ 2. The equation 0(i?)a; =/(<). 



§ 2-1. The solidionfor definite initial data obtained from the equation itself. 



If we replace (10) by 



i>{D')x-f{t') = {), (16) 



and follow treatment precisely similar to that in § (1'3), the solution (15) 

 is obviously replaced by 



^ c^'dX - 4>{D)~ ^{X) 

 7J ?'(X)L D-X '• 

 dX f« 



i-i^XXt ■■ 



J 1=0 



^c\m\<r'"'^''^''- 



(17) 



