The author refines results on Jordan (triple) homomorphisms proved in an earlier paper [Part I, J. Algebra 127, 218-228 (1989; Zbl 0691.16040
)]. An additive mapping of rings T: $R\to S$ satisfying $T(ab+ba)=T(a)T(b)+T(b)T(a)$ is a Jordan homomorphism, and is a Jordan triple homomorphism if $T(aba)=T(a)T(b)T(a)$. The results assume that S is 2-torsion free and that the annihilator of any ideal in S is a direct summand of S. The first theorem shows that when T is a Jordan homomorphism, then $R=I+J$ for ideals of R with $I\cap J=Ker T$, S is the direct sum of ideals T(I) and T(J), and T is an associative homomorphism on I and an antihomomorphism on J. For the second main theorem, $R\sp 2=R$ and T is a Jordan triple homomorphism. In this case there are four ideals $U\sb 1,...,U\sb 4$ of R whose sum is R and whose pairwise intersections are all Ker T, so that S is the direct sum of the ideals $T(U\sb i)$, and T acting on each $U\sb i$ is either an associative homomorphism, antihomomorphism, or the negative of one of these.