we operate on it by ∂/∂xσ. Now,
∂²/∂xα∂xσ (gσβΓμβα)
= -½ ∂²/∂xα∂xσ [gσβ gαλ(∂gμλ/∂xβ
+ ∂gβλ/∂xμ - ∂gμβ/∂xλ)].
The first and the third member of the round bracket lead to expressions which cancel one another, as can be easily seen by interchanging the summation-indices α, and σ, on the one hand, and β and λ, on the other.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂xα∂xσ (gσβγμβα)
= ½ ∂³gαβ/∂xσ∂xβ∂xμ
The second member of the expression on the left-hand side of (52a) leads first to
- ½ ∂²/∂xα∂xμ (gλβΓλβα) or