  A new method of reaction analysis for proton elastic scattering is proposed,
combining systematic analyses of nuclear structure and reactions in a series of
isotopes. This method is applied to $^{204}$Pb$(p,p)$, $^{206}$Pb$(p,p)$, and
$^{208}$Pb$(p,p)$ at $E_p=295$ MeV to obtain improved neutron densities and
radii from experimental cross section data. The reaction calculation is
performed according to the relativistic impulse approximation using the
modified Murdock and Horowitz model with density-dependent effective $NN$
interaction. Analysis of Pb isotopes is performed using theoretical densities
given by the relativistic Hartree-Bogoliubov and nonrelativistic Skyrme
Hartree-Fock-Bogoliubov calculations and the experimental density extracted
from the $(p,p)$ data at 295 MeV. The isotopic ratios ($R(\sigma)$) of the
$^{A}$Pb$(p,p)$-to-$^{206}$Pb$(p,p)$ cross sections are analyzed in connection
with the isotopic differences ($D(\rho_n)$ and $D(r_n)$) in neutron densities
$(\rho_n(r))$ and radii $(r_n)$ between $^A$Pb and $^{208}$Pb. A hole-model
analysis is performed, assuming $^{204}$Pb and $^{206}$Pb as a $^{208}$Pb core
and neutron holes to clarify the one-to-one correspondence between isotopic
structure properties such as $D(\rho_n)$ and $D(r_n)$ and the isotopic
cross-section ratio. By fitting $R(\sigma)$, the values of $D(\rho_n)$ and
$D(r_n)$ are extracted from the experimental $(p,p)$ cross section data with
less uncertainty of the structure and reaction models. The isotopic neutron
radius difference between $^{206}$Pb and $^{208}$Pb was obtained as
$D(r_n)=-0.012$ fm with an acceptable range of $D(r_n)=-0.03 \sim -0.006$ fm.
