  We study equations over torsion-free groups in terms of their `t-shape' (the
occurences of the variable t in the equation). A t-shape is good if any
equation with that shape has a solution. It is an outstanding conjecture that
all t-shapes are good. In [Klyachko's methods and the solution of equations
over torsion-free groups, l'Enseign. Maths. 42 (1996) 49--74] we proved the
conjecture for a large class of t-shapes called amenable. In [Tesselations of
S^2 and equations over torsion-free groups, Proc. Edinburgh Maths. Soc. 38
(1995) 485--493] Clifford and Goldstein characterised a class of good t-shapes
using a transformation on t-shapes called the Magnus derivative. In this note
we introduce an inverse transformation called blowing up. Amenability can be
defined using blowing up; moreover the connection with differentiation gives a
useful characterisation and implies that the class of amenable t-shapes is
strictly larger than the class considered by Clifford and Goldstein.
