') (9) 



1177 



3. ., 



A = aV(zl a')z' — -zV(ai z')a' = a 7a'-z7z' ') . . (6) 

 or shorter: 



3 . . , 



'Vv =: Vv — A 1 V 1 



3. . ') (7) 



'Vv'= Vv' + v'^. A . ' 



3. ., 



The affiiior A is a simuUaiieoiis covariaiil of '1 and 'j, symme- 

 trical witli respect to the two tirst ideal factors, V v= V v 

 being independent of the fundamental tensor'). In the same way 

 we have : 



'dp = dp . . ^. (8) 



3.., 

 'd\ :=: d\ — (ix'l A \ V 

 3 . . , 



'Jv'=(/v' f c/x'v'! A 



For every co\ariant affinor v = Vj . . . v^, we find, taking into 



consideration : 



p 



v = (a'i v/)vi....v/--i aVi-|-i....v^, ') (10) 



and the equation : 



i' I' 



V -^ a^ .... ay, (a,'.v J .... (a^'.v,.) = a, . . . a^, a^,' ... a/ /.' v . (10a) 



which follows from (10) and making use of (7): 



'7v — 'i ('Vv,)i a 'Vj .... v._, a v-^^ . ■ • • v^, = 



i 



= -7y-"i\k^. V,.) 1 a' V,. . . . y;_, av,._^j • • • • v, = (1 1) 



= 7.v-p'(A 1 a,)i a' a,....a,_i a a,_^, .... a^, a^, • • • . a, /;vj 

 and therefore : 



I' i> ll,-,/' 3... I V 



'dv=zdx—dx'\ \:^(X U/)U'a,....a,_iaa,_^i. ..a^,a^;....a,' /.'v. (12) 



This is the proof for a covariant aflinor. For contravariant and 

 mixed atfinors the proof can be given in the same way. 

 For the special case that : 



M = ^e^e, (13) 



^) A mixed affinor will be indicated by an index of points and commas referring 

 to the place of the covariant and contravariant ideal vectors. 

 2) Gomp. A.R. p. 89 form. (1036). 

 ') Gomp. A. R. p. 55. 

 *) Gomp. A.R. p. 89 form. (108«). 

 ') Gomp. A. R. p. 54 form. (74). 



