Observations of 0957+561 with the Einstein High Resolution Imager (HRI) in 1979 May and 1979 November and with the ROSAT HRI in 1991 May and 1992 October reveal large variations in the X-ray fluxes for images A and B which significantly differ from simultaneously observed changes of the corresponding optical continuum emission. Most striking was the apparent increase by over fivefold in the flux of image B from the late 1970's to the early 1990's. Further, in the 1990's the X-ray flux for image A increased by a factor of 1.7 +/- 0.3 and that of B by a factor of 1.9 +/- 0.2 between the two ROSAT observations (separated by 540 days), whereas optical measurements showed nearly no changes for A and B between these epochs. No significant changes in X-ray emission were observed on timescales of hours to days. If we adopt 1.5 yr as the difference between the propagation times from the quasar to us via the two images (''time delay''), then the ratio of the X-ray flux of image B for the 1992 October epoch to that of A for the 1991 May epoch is 2.7 +/- 0.4. This ratio is significantly greater than the ratio of 1.05 +/- 0.03 observed in the optical R band, 0.75 in the broad line region (BLR), and 0.76 +/- 0.03 in the radio (VLBI lambda 13 cm core) for approximately the same epoch. The wavelength dependence in the ratio of the fluxes of the two images suggests that either microlensing may be significant for the X-ray band or the time delay is substantially different from 1.5 yr (and the intrinsic variation of the quasar emission were significant within an interval comparable to the uncertainty of the time delay), or both. A ROSAT Position Sensitive Proportional Counter (PSPC) spectrum of 0957+561 is fitted well with a power law of energy index 1.1 +/- 0.2 plus a Raymond-Smith thermal plasma model with a temperature of 0.22 +/- 0.17 keV in the source frame. The estimated unmagnified X-ray luminosity of the thermal component is 3 x 10(44) ergs s(-1) (if the magnification of the thermal component by the lens is 14). This thermal emission may be a result of a strong cooling flow in the vicinity of the quasar. There is no evidence for excess absorption above that attributed to our galaxy for the relatively high-redshift quasar. We place an upper limit of 6(-4)(+3) x 10(20) atoms cm(-2) (90% confidence) on absorption at the redshift z = 1.391 of a known damped Lyman-alpha absorber and provide upper and lower bounds of 30 h(50)(-1) kpc and 0.2 h(50)(-1) kpc, respectively, for the spatial extent of this H I region. For the cluster of galaxies around the primary lensing galaxy, G1, the 3 sigma upper limit on the X-ray luminosity within a 0.5 Mpc radius over the energy range 2-10 keV (PSPC frame) is 4.1 x 10(44) ergs s(-1) for a cluster core radius of 0.1 Mpc and 1.7 x 10(44) ergs s(-1) for a core radius of 0.25 Mpc. Similarly the limit on the X-ray luminosity of a cluster around the galaxy G5 is found to be 2.8 x 10(43) ergs s(-1). Based on the relation of cluster X-ray luminosity to gas temperature and system mass, we estimate the 3 sigma upper limits on the mass of the clusters that contain the galaxies G1 and G5 within 1 Mpc to be 1.5 x 10(14) and 1 x 10(14) M., respectively.
