  A partial wave analysis is presented of $J/\psi \to \phi \pi ^+\pi ^-$ and
$\phi K^+K^-$ from a sample of 58M $J/\psi$ events in the BES II detector. The
$f_0(980)$ is observed clearly in both sets of data, and parameters of the
Flatt\' e formula are determined accurately: $M = 965 \pm 8$ (stat) $\pm 6$
(syst) MeV/c$^2$, $g_1 = 165 \pm 10 \pm 15 $ MeV/c$^2$, $g_2/g_1 = 4.21 \pm
0.25 \pm 0.21$. The $\phi \pi \pi$ data also exhibit a strong $\pi \pi$ peak
centred at $M = 1335$ MeV/c$^2$. It may be fitted with $f_2(1270)$ and a
dominant $0^+$ signal made from $f_0(1370)$ interfering with a smaller
$f_0(1500)$ component. There is evidence that the $f_0(1370)$ signal is
resonant, from interference with $f_2(1270)$. There is also a state in $\pi
\pi$ with $M = 1790 ^{+40}_{-30}$ MeV/c$^2$ and $\Gamma = 270 ^{+60}_{-30}$
MeV/c$^2$; spin 0 is preferred over spin 2. This state, $f_0(1790)$, is
distinct from $f_0(1710)$. The $\phi K\bar K$ data contain a strong peak due to
$f_2'(1525)$. A shoulder on its upper side may be fitted by interference
between $f_0(1500)$ and $f_0(1710)$.
