  We study the thermodynamics of the delta-chain with competing ferro- and
antiferromagnetic interactions in an external magnetic field which generalizes
the field-free case studied previously. This model plays an important role for
the recently synthesized compound Fe$_{10}$Gd$_{10}$ which is nearly quantum
critical. The classical version of the model is solved exactly and explicit
analytical results for the low-temperature thermodynamics are obtained. The
spin-$s$ quantum model is studied using exact diagonalization and
finite-temperature Lanzos techniques. Particular attention is focused on the
magnetization and the susceptibility. The magnetization of the classical model
in the ferromagnetic part of the phase diagram defines the universal scaling
function which is valid for the quantum model. The dependence of the
susceptibility on the spin quantum number $s$ at the critical point between the
ferro- and ferrimagnetic phases is studied and the relation to
Fe$_{10}$Gd$_{10}$ is discussed.
