We report measurements of Al-26, Be-10, Ca-41, and Cl-36 in the silicate and metal phases of 11 mesosiderites, including several specimens each of Budulan and Estherville, of the brecciated meteorite Bencubbin, and of the iron meteorite Udei Station. Average production rate ratios (atom/atom) for metal phase samples from Estherville and Budulan are Al-26/Be-10 = 0.77 +/- 0.02; Cl-36/Be-10 = 5.3 +/- 0.2. For a larger set of meteorites that includes iron meteorites and other mesosiderites, we find Al-26/Be-10 = 0.72 +/- 0.01 and Cl-36/Be-10 = 4.5 +/- 0.2. The average Ca-41/Cl-36 production rate ratio is 1.10 +/- 0.04 for metal separates from Estherville and four small iron falls. The Ca-41 activities in dpm/(kg Ca) of various silicate separates from Budulan and Estherville span nearly a factor of 4, from <400 to >1600, indicating preatmospheric radii of >30 cm. After allowance for composition, the activities of Al-26 and Be-10 (dpm/kg silicate) are similar to values measured in most ordinary chondrites and appear to depend only weakly on bulk Fe content. Unless shielding effects are larger than suggested by the Cl-36 and Ca-41 activities of the metal phases, matrix effects are unimportant for Be-10 and minor for Al-26. Noble gas concentrations and isotopic abundances are reported for samples of Barea, Emery, Mincy, Morristown, and Marjalahti. New estimates of Cl-36/Ar-36 exposure ages for the metal phases agree well with published values. Neon-21 production rates for mesosiderite silicates calculated fi-om these ages and from measured Ne-21 contents are consistently higher than predicted for L chondrites despite the fact that the mesosiderite silicates have lower Mg contents than L chondrites. We suggest that the elevation of the Ne-21 production rate in mesosiderite silicates reflects a "matrix effect," that is, the influence of the higher Fe content of mesosiderites, which acts to enhance the flux of low-energy secondary particles and hence the Ne-21 production from Mg. As Be-10 production is relatively insensitive to this matrix effect, Be-10/Ne-21 ages give erroneously low production rates and high exposure ages. By coincidence, standard Ne-22/Ne-21 based "shielding" corrections give fairly reliable Ne-21 production rates in the mesosiderite silicates.