Functions of the Form Y{z + x). 285 



tlie ?/th term being represented by 



< « alternations of /and -y- >Yzdt dt' dt" he. 



But since all the limits are determined by the outermost 

 limits and x, we may take away the accents from t' i", &c., 

 and the notation of the limits from all the signs of integration 

 but the first, and write the development thus: 



Since we have not determined whether x shall or shall not 

 be a function of z, we do not know whether or not we may 

 invert the order of differentiation and integration. In the first 

 term, however, the function under the sign of differentiation 

 does not contain t explicitly, and therefore it becomes 



d /*^ d 



-rFz/ dt or -y- Fz.x. 

 dz ^ Q dz 



Now Laplace's theorem is a case in which x is a function 

 of 2-, such that we may change the order of integrations and 

 differentiations in all but \k\Qjirst term. For let 



dz - ^' 

 then if n be a whole number but not cipher, 



y y ^z . V\ dt =r U/z.t'' + ^z. «/«- ' ^) dt 

 = ^'z ^ + n^z . I -i-t-'dt 



and 



=^^Tn""^'-"^"' 



nnd generally 



- / •\>zt".dt= -7- '-v:^' — r-: 

 zJ^^ dz 71 + 1 



n+\ dz 



\x 



d"* /^' / ' d"^ 



j^J i,x,t"dtt,\%o=J ^^Jz.V'dt; 



