36 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
Figure 1. The first few graphs in the expansion of W (P, I).
The variable a U is placed at each external edge.
Contracting these two elements yields the weight wγ,φ(P, I)(a1,...,an).
Define a function
wγ(P, I) O(U )
by
wγ(P, I)(a) = wγ,φ(P, I)(a,...,a)
where φ is any ordering of the set of tails of γ. Note that wγ(P, I) is
homogeneous of degree n, and has the property that, for all a1,...,an U,

∂a1
· · ·

∂an
wγ(P, I) =
φ:{1,...,n}∼T
= (γ)
wγ,φ(P, I)(a1,...,an)
where the sum is over ways of ordering the set of tails of γ.
Let vi,k denote the graph with one vertex of genus i and valency k, and
with no internal edges. Then our definition implies that
wvi,k (P, I) = k!Ii,k.
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