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Take transverse immersions f from a disjoint unin of the three 4-spheres$S^4_1$, $S^4_2$, and $S^4_3$ into $S^6$ with the following properties:  (1) The restriction of $f$ to $S^4_i$ is an embedding,  (2) The intersection of $f(S^4_i)$ and $f(S^4_j)$ is not empty and connected,  (3)The intersection among $f(S^4_1)$, $f(S^4_2)$, and $f(S^4_3)$ is notempty.  Then we obtain three surface-links $L_i=(S^4_i\cap S^4_j, S^4_i\cap S^4_k)$in $S^4_i$, where $(i,j,k)=(1,2,3), (2,3,1), (3,1,2).$ We prove that, we havethe equality $\beta(L_1)+\beta(L_2)+\beta(L_3)=0$, where $\beta(L_i)$ is theSato-Levine invariant of $L_i$, if all $L_i$ are semi-boundary links.

The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant