After a good fair bit of head-scratching about getting the hole pattern correct for the globe just right, getting close but not quite right (see the first photo!), I finally figured it out. 

 This necessitated a whole new rig for the hole drilling. The first one that I made was too wobbly and imprecise. The new one is a turntable mounted so that it can rotate under the drill press. Much better! I have the two new globes made up for the screens, now, so I can work away at this for the next good while. I tested it out today and so far so good.