298 



WILSON AND MOORE. 



The second term here is 



V Q 

 I 



The first term may be written as 



'-'ilpq'Ysfyrp'y tqK-^ ij) 



Sv q 



' ijuvmnpql-^ uv(^ui(^v jC piy s i^t' TmiTni 



r t 

 n 



r t 

 — ^ X 



\r t 



m 



Hence 



'^ijiXij) — 7sy = Zm 



OXrOXt 



and in hke manner, 



2^ij{Xij) - — ^ — yri = ^T, 



X 



r t 



V .\fSP^ 



m3 1 ( ^^ipq'Ysi'YTp'y tq\-^mi) \ \ 



m ) \( m 



dxsdxt 



Hence 



^rst — 



dXrs 



dxt 





r t) is t 



I m ) ( m 



P 9 

 m 



(36) 



transforms covariantly as of order three. 

 We may generaHze to the next higher order as 



■^rstu — 



dX, 



rst 



dxu 



( m ) ( m ) I 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 



dXrdXs "^ dXm ( W \ 



