Have you checked the Airline website that your flying with?
The allowance is 23kgs we usually take 3 cases with us and we aim to have no more than 18kgs in each case and with BA so far not a problem as they know from our e-ticket where we are going but certainly with BA on this flight it can be a 1 + 2 seating arrangement with the smaller plane they use from time to time but you might not be with BA as Wilba asks.
As airline baggage allowances vary depending upon sectors flown, level of ticket purchased and passenger status within the frequent flyer programme, as Wilba said, check with your airline.
If you are flying BA and you think the 23kg each will be a problem, they are quite reasonable for purchasing an extra case for your allowance. For example from Edinburgh to London it is only £20 for another 23 kilos if you buy online. This compares well to other carriers, for example, BMI charges £8 per kilo on the same route.
Currently only Flybe appear to cover Edin/Southampton and their allowance is 20kgs in the hold and 10kgs in the cabin unless you have Economy Plus which will allow 30kgs in the hold and 10kgs in the cabin.
Blyth is correct, Flybe are the airline and the allowance is 20KG. You pay extra for bags - £7.99 per bag each way if you are booking online. Excess baggage fees are £10 per kilo and they state that excess baggage is carried standby - in other words only if there is space available. However, a couple of kilos over on an already booked bag would make it very unlikely that your bag will be off-loaded. If you book Economy Plus it will cost you an awful more than normal Economy but your 30KG of baggage is included in the fare.