Solution
= Solution
Take the dumbbell axis to be the $z$ axis, with its midpoint at $z=0$. The right-hand <sphere> has centre $z=\alpha a$ and <radius> $R=\beta a$, while its junction with the <cylinder> is at $z=\gamma a$. Put $d=(\alpha-\gamma)a$, so $0<d<R$. The <Pythagorean theorem> in a meridional cross-section gives the handle <radius> $H$ from $H^2+d^2=R^2$. Thus
$$
\boxed{H=a\sqrt{\beta^2-(\alpha-\gamma)^2}.}
$$