The .310 Remington was a small brass-cased rimfire shotgun cartridge made by Remington for a miniature skeet shooting game. Shell length was only 1-1/16th (1.0625) inches.
According to Frank Barnes' Cartridges of the World (14th Edition), the shotgun it was fired in was a bolt-action that was used to shoot small clay pigeons. However, "historian" at the Cartridge Collectors forum notes
There have been other very similar products on the market, introduced by various companies. Some examples are Mossberg's "Targo" and Winchester's "Wingo" - though the latter was to be played indoors. Another difference is that Mossberg's "Targo!" game used typical .22LR shotshells and a smooth bore .22 "shotgun" that could also accept a rifled adapter for shooting standard ammunition, similar to rifled chokes available for smoothbore shotguns.
Information is scarce on this ammunition, so it's not terribly clear how much shot a shell contained, or at what velocity they would fire. Let's do some guesswork..?
Let's assume the inside diameter of the casing is .310 caliber. A .32 ACP's overall length is actually a little shorter than the length of Remington's .310 Skeet shells. A number of manufacturers offer shotshells in various calibers, though CCI is probably most known for this. Their typical projectile weight is about half what is typical for a caliber. For example, their .45 ACP shotshells only hold about 120 grains of shot, even though a classic heavy loading for the .45 ACP would be a 230 grain FMJ bullet.
Despite being a smaller diameter than a .32 ACP, these shells are also longer, so let's assume similar overall performance. Typical FMJ bullet weight for .32 ACP is 71 grains, so half of that is 35.5, so let's round up to 40 grains of shot, since CCI's .22 shotshells actually are loaded with 31 grains of shot at 1000 feet per second, which is more than the half weight we see in some of their other shells they sell. It should also be noted that their .45 ACP shotshells fire at 1,100 feet per second, and their 9mm shotshells fire as fast as 1,450. Let's just use 1,000 as a fair guestimate.
So that's 40 grains of shot, at 1,000 feet per second, for approximately 88.84 ft lbs of energy.
It could easily be more, or maybe even less. Remember, these shells and their equipment were marketed to golf ranges and similar venues and appear to be meant for relatively close ranges for miniature clay pigeons, so it's unlikely Remington's .310 Skeet ammunition packs any serious energy.