298 



WILSON AND MOORE. 



The second term here is 



^ipqy»iyrpytq^ii{Xij){a^'^^){ 



I 



= ^iipqlsilrpytqiXiM } . > 



The first term mav be written as 



'l]uvmnpq 



l^uvCuif^vjCpl'YsiCl y miy nl 



r t 

 .P. 





r t 

 n 





m 



Hence 



OXrOXt 



Y ^'' ^( _ V ^ (Y \l S'P ^ 



(m) \( m 



and in Hke manner, 



2ty(Xj,-) '- — Jri = ^n 



dXsdxt 



X 



- ' — S;p57r;73p7«9(-3r;m)( ' 



I 



Hence 



Y _ dXrs _ y 



dxt 



m 



I m ) I m 



m 



(36) 



transforms covariantly as of order three. 



We may generaHze to the next higher order as 



■^rstu 



dX 



rst 



dxu 



Xmst ') r + Xrmt ] f + Xr. 



( m 



m 



t u 



m 



and so on. These derivatives of higher order may also be written 

 neatly by using matrical notation, but we shall carry that method no 

 further. 



A particular case of interest is the successive covariant derivatives of 

 a function F. The first is merely the set Xr = dF/dxr as shown above 

 (§14); the second is 



„ _ fF__ dl\r s] 

 dx,dxs oXm ( m ) 



