It seemed that Psexec would solve my problem, but I had the common Access Denied error.

So, after fiddling with it a bit and keeping straight which computer was "active" when the file would run, I wound up taking eccentricfusion's advice and set up a scheduled task. So the desired little program runs on the proper machine and all is well. So far.

Still have to figure out the problem with Psexec. Something to do with ADMIN$ shares and so forth. The explanations given seem reasonable, but you have to already know what they're talking about before you can understand what they're talking about. Reminds me of my early efforts to get Linux up and running ... I was essentially told that, if I didn't already know all this stuff, I wasn't worthy of running Linux.