what is Arthur really hoping for with the marrige?

Birling tells Gerald that he is “just the kind of son-in-law I always wanted” and that Gerald and Sheila will make each other happy. He also makes clear, none too subtly, that he has ambitions for Crofts Limited and Birling and Company (the smaller of the two firms), though they are currently competitors, to work together at some point in the future, as a result of this marriage.