TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 33 



then the limits of t' will be and ( — a — w), and we have 



where, considering that 



(_<)»/+! _ (-i)»ir _ Or l) w 



V+ 1 ~mn.(t-w)~ i e/+1 ' 



we have 



By (I. ) the second member will be 



"»=(-!)" ■ g+g^+i . where A- -a. 

 But we have 



and as the second factor on the right is 



(-ir.(- 7 +/3-^+ir +i , 



(- 1)-( + / 8- 7 + !)«/+!, 



namely, 



we have first 



Then 

 hence 



^ 7 W (a + /3- 7 + l) w/+1 



( 7 + w yi/+i = ( 7+w )'.-«»/+ix( y +w+^-wf/+ , 

 1 = (y+t 1 ) w ' +1 _y w i +1 



(7+w)'i-»/+i - ( 7 + w ,yi/+i yi/+i' 



the third member being deduced from 



y 1 +« I /+i_y 1 /+i^^.^« > /+i_ 7W /+i^_l_ w y I / + i 



As 



(-irx(-i) w =+i, 



we have now 



Replacing t x = - a, and introducing U M in the last expression of 



\ rye ' 



we get 



m^ F /a^^V._ (y-/g)- a/+1 F f a,/3,(e-a) ) 



U; 'Vy.ei y- a / +1 I! l(a + /3-y+l), e i ' 



From this we might draw several other combinations. We will only notice the one 

 which we shall make use of further on. 



