When n > 2, these equations are reducible by means of (36.7) to Consequently in applying the results of § 8 we denote by equations (36.3) and proceed with these equations to get the sequence … of derived sets. If we multiply (36.5) by and sum for α and we obtain Since (32.16) follows formally from we have from (35.12) and (36.6) We observe from (35.12) and (32.10) that is of the same form in as it is in, when the tensor is symmetric. In this notation equations (35.17) may be written i11 the form Where denotes the projective derivative of, that is, the covariant derivative with respect to the we remark that the projective derivative of a tensor is not in general a tensor. If we differentiate equations (36.3) with respect to the resulting equations are reducible by means of (36.2) and (36.3) to Which is satisfied identically because of (36.1) and If we denote by the cofactor of in the jacobian divided by the jacobian, we have from (35.6)
From the preceding results it follows that the conditions of integrability of these equations are These equations together with the functional relation (35.6) must admit a solution for given expressions of and in the and respectively, if the two projective connections so defined are to be equivalent.
The equivalence of projective connections.
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