Only two clauses to go, and they're about being stroppy, so perhaps we'll do them both at once. W00t- stroppy-fest!
If any differences shall arise between the Author and the Publishers touching the meaning of this Agreement or the rights and liabilities of the parties thereto, the same shall be referred to the arbitration of two persons (one to be named by each party) or their mutually agreed umpire, in accordance with the provisions of the Arbitration Act 1950, or any amending or substitute statute for the time being in force.
This means if you disagree about agreeing you have to agree on how to stop disagreeing. That's no better, is it? It means your understanding of the contract is not the same as theirs, and you discover this because you argue about something, can each pick someone to argue for you - like a court case but cheaper. Don't pick your partner/neighbour/best friend. Pick someone who knows what they're talking about. (Of course, if your partner/neighbour/best friend is a lawyer specialising in publishing law, use them anyway.) You could ask the Society of Authors to represent you in the fight. Note that it will not be a physical fight, so someone big, strong, fat as sumo wrestler or trained in martial arts is not necessarily the best choice - unless things go badly wrong, or they also happen to know about publishing law. In which case, please send me their email address.
I'm not sure who might be a 'mutually agreed umpire', but I suppose depending on the nature of the dispute you might agree on a lawyer who is not representing either side. But lawyers are expensive - make sure you don't stand to lose more than you could gain if you follow the legal route. If it's a small and naive publisher you might get the to agree to go to the Society of Authors as umpire, but that word 'Authors' in the name might alert them to the possibility that the Society might take your part.
What to disagree about (if you're looking for a chance to use this clause)? It suggests the meaning of the contract, the rights and liabilities. Rights are the most likely cause of dispute as new rights emerge from nowhere without warning, rather like the discredited spontaneous generation of worms from mud. If you signed a contract in 1995, app rights were not even a glimmer in your publisher's eye so you might now argue about those. In 2020, are you going to be arguing about holographic rights? The best way to avoid this is to withhold all non-specified and future rights from the agreement, or at the very least put a time limit on them - if additional rights are not exploited in a year they revert, for example. You don't want the publisher holding on to rights they are not going to use when you could exploit them. But now we're going back through the contract to the rights clauses, so perhaps you should take another look at those to make sure you won't need this arbitration clause.