relator. In its modern form, the inductive step in the classical proof expresses a one-relator group $G$ as an HNN-extension of a one-relator base group $G^*$ where the edge subgroups are Magnus subgroups of $G^*$. Thus Magnus subgroups play a central role in this approach to the theory of one-relator groups.

In a previous article [@C], we determined the form of the intersection of two Magnus subgroups. The precise statement is:

\[thm1\] Let $G = \langle X : r=1 \rangle$, where $r$ is cyclically reduced, be a one-relator group and let $M=F(S),N=F(T)$ be Magnus subgroups of $G$. If $M\cap N$ is distinct from $F(S\cap T)$, then $M\cap N$ is the free product of $F(S\cap T)$ and an infinite cycle.

In the present article we examine the intersection of conjugates of two Magnus subgroups, and it suffices to deal with the case of an intersection of the form $gMg^{-1}\cap N$, where $M=F(S),N=F(T)$. A simple and obvious argument shows that if $g\in NM$, then $gMg^{-1}\cap N$ is just a conjugate of $M\cap N$ by an element of $N$ and in particular is isomorphic to $M\cap N$. Our main conclusion deals with the alternative case.

\[thm2\] Let $G = \langle X : r=1 