6 9 2 D O C U M E N T 4 1 8 C A L C U L A T I O N S I N D I A R Y the form . [26]Einstein implicitly introduces the notation of for the determinant of and rewrites the first two terms of the preceding line (the factor A of the preceding note), divided by . The complete expression would read , where the last four terms are indicated in the manuscript by dots. [27]Einstein first contracts [eq. 11] with , obtaining [eq. 12], after introducing the notation and . He then contracts [eq. 11] with , obtaining [eq. 13]. Eqs. [12] and [13], after further simplification, imply [eq. 14], which, going back to [eq. 11], implies [eq. 15]. A similar calculation is done on [p. 11]. [28]The third term in [eq. 12] should read . [29]On the problem of asymmetric connections, see Doc. 417, note 4, and Einstein 1923e (Doc. 425), note 8. See also the similar calculation on [p. 42v]. [30]The following calculation recapitulates the derivation of the Riemann curvature tensor by par- allel transport around a closed loop. The derivation closely follows the treatment in Einstein 1922c, pp. 49–50 (Vol. 7, Doc. 71, pp. 548–549). This derivation had earlier been the subject of correspon- dence between Einstein and Paul Dienes, who had challenged the mathematical rigor of the derivation as done by Weyl and Eddington. In his response, Einstein had sent copies of the relevant two pages to Dienes, claiming that the derivation is rigorous if done carefully (see Doc. 338). [31]The left-hand side of this equation should read . [32]The second term should be preceded by an integral sign. [33]In Einstein 1922c, p. 49 (Vol. 7, Doc. 71, p. 548), this step was justified as follows: “Subtracting from the integrand, we obtain . This skew-symmetrical tensor of the second rank, , characterizes the surface element bounded by the curve in magnitude and position” (“Zieht man vom Integranden ab, so erhält man . Dieser antisym- metrische Tensor zweiten Ranges charakterisiert das durch die Linie gelegte Flächenelement nach Größe und Lage”). [34]The expression is missing a factor of . [35]The left-hand side of this equation should be . [36]Einstein starts another calculation, along the same lines, of the variational integral , as he had done above on [p. 48v] see notes 24ff. [37]The partial derivatives of in the last two terms should read or . [38] should be . [39]The following calculation is similar to an earlier one on [p. 8] see note 27. AδΓiακdτ 0= D Rik ≡ Rik D ∂Riκ ∂xα ----------- Riκ--------------------- 1 D ∂ D ∂xα ∂Riβ ∂xβ -----------δα κ – Riβ---------------------δα 1 D ∂ D ∂xβ κ – RiβΓβα κ RκβΓβα i RστΓστδα i κ – RiκΓαβ β – + + + Riκ Di Riσ σ ≡ Eα RστRστ α ≡ δακ 1 2 --RiκDκδα - i ΔAμ 1 2 --d(ξαξβ) - 1 2°( -- - ξαdξβ ξβdξα) – fαβ 1 2 --d(ξαξβ) - 1 2°( -- - ξαdξβ ξβdξα) – fαβ Aα · Rαβ δ RαβδRαβdτ 0= D 1 D ------------------- - ∂ D ∂xσ - ∂lg D ∂xσ ---------------- - RαβR,αβ σ RαβRαβ σ