When it comes to Riihimaki/Riihimaen, px is the expert, and has helped me several times in the past. :thup:
As for the name difference, as I understand it, the factory was originally called Riihimaki (after the town of Riihimaki in Finland) and was later renamed Riihimaen Lasi Oy in the 30's. Both names are equally valid I think, but I guess Riihimaen would be the most correct? Maybe Riihimaki is used a lot because that is what ebay calls the category for this glass. It's also the name I hear most often when picking up this glass at antique fairs etc.
petet63 - much better pics! I think it is safe to say your vase is the 1378.