Hypnotism, in the book and Television series, is the power that The Headmaster uses to take control of his subjects minds and force them to follow his commands. He hypnotises people by taking off his glasses and convincing people to look into his eyes whilst putting them under by his soothing voice.
After he has put them under his control, he is able to keep them under for a long time and to even program them to with trigger phrases when people ask them certain questions. An example of this is pupils of St Champions who are programmed to respond to any questions about their Headmaster with "I think the Headmaster is a marvellous man and this is the best school I've ever been to".
Whilst he is able to control most people with his powers there is however a small amount of people who are immune to his powers like the members of SPLAT, with the exception of Dinah. These people are a problem to the Headmaster plans but are in reality a small number eg. six students out of an entire school of Pupils.
By the time of "The Prime Minister's Brain" he has invented an Octopus design which he uses to control people who look at it. He uses the Octopuses in the Junior Computer Brain Of the Year competition, where he incorporates into the Octopus Dare game, cards and screens inside the Saracen Tower. This design is more effective than his ordinary as it is able to take control of Lloyd, Mandy and Ian.
History of usages
When we first meet the Headmaster, he is the Headmaster of St Champions school where he has hypnotised all the pupils and staff and uses the School Prefects to maintain order.