What is this hose for?

[BigRed]

While changing out my differential oil today, I noticed a hose coming out of the top of the diff and it goes up to a clamp on the frame. The hose is plugged right past that clamp on the frame. At first, I thought this might be a fill line, but I don't think so. The hose was split so I replaced it. What is this hose for?

Also, I refilled my diff through the plug on the passenger side of the diff case. I basically filled it through that hole until it started seaping out. Is that as full as I need to fill it?

Thanks,