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XXXY. The Identical Relations in Einstein' 's Theory. 

 By A. E. Harward*. 



THE March number of the Philosophical Magazine con- 

 tains an interesting proof of the identity 



by Dr. G. B. Jeffery. 



Apparently it is not generally known that this identity is 

 & special case of a more general theorem which can be very 

 easily proved. I discovered the general theorem for myself, 

 but I can hardly believe that it has not been discovered 

 before. 



The theorem is 



(BpyoPjr + (B^/)„ + (BptS)* =0. . . (1) 



This identity can be verified in a rather laborious manner 

 by forming the covariant derivative of B^^P, but it can be 

 more easily proved as follows : — 

 The identity 



Aju, V( j — Aju, <Tv — j5fivo p Ap .... (2 J 



can be easily generalized so as to apply to the case where 

 instead of the vector A^ we have a tensor of any order ; thus 



Aftv, or — Apy, tg == -t>ju<7r Apv ~T Dyur -^H-P' 



This is proved in the same way as (2). 

 Now, if Afx be any covariant vector, then 



(Aju, vot — Aju, vt(t) -f (Aju, arv — &p., <rvr) + (^/z, rva A^ ; rav) 



= (A^, va — A^ ; cv)t-{- (A^, ar — Ap, rtrjv + (A^, rv — A^ V r)a\ 



J3 JLt(Tr |0 A.p^ v -\- D V(TT P Ap ( p -4 lJfiTV ^-p, <T + *5<TTV Ayu, p 

 + ^fxva P Ap, T + ¥>TV(T P Ap 5 p 



= (Bp Va P Ap)r + ( B^ T P A p ) v + (B Mr / Ap) a . 

 * Communicated by the Author. 



