In chapter 6, Samneric report that they have seen the beast. Because of their convincing testimony, the boys now believe that the beast is real. Jack is excited to go find the beast, saying, "This'll be a real hunt. Who'll come?" He sees this as an opportunity to demonstrate his leadership. He is still jealous of Ralph and feels competitive towards him. Perhaps he senses weakness in Ralph and believes this will be his moment to wrest power away from him. He taunts Ralph, asking him if he is frightened. Ralph replies honestly, "'Course I'm frightened. Who wouldn't be?" Ralph is secure enough in his leadership position to admit his fear, but Jack hears opportunity knocking. His confidence has been boosted by recently killing a pig; he believes he can kill this beast as well.
Ralph's motives for wanting to find the beast are completely different. The fear of the beast has resulted in Samneric leaving their post as keepers of the signal fire. The fire is their only plausible chance of rescue, and Ralph tenaciously clings to the fire as his primary purpose. In order to get the signal fire lit again, they must be able to climb the mountain without fear. Whatever fear Ralph has of the beast, he has more fear of not being able to have the signal fire emitting its potentially life-saving smoke. So while Jack's motives for seeking the beast are selfish, Ralph's motives are altruistic.
During the meeting to discuss finding the beast, Jack ridicules Piggy and his claim of the conch. He says that the conch is not needed anymore. In his opinion, free speech and democracy have no value. He states, "We know who ought to say things. What good did Simon do speaking, or Bill, or Walter? It's time some people knew they've got to keep quiet and leave deciding things to the rest of us." Jack shows that he prefers an oligarchy at least, and if he can achieve it, a dictatorship. He believes that those with power should make the decisions according to their own judgment.
At this point Ralph is able to reassert his authority by asking the boys, "Don't any of you want to be rescued?" This causes "a violent swing to Ralph's side, and the crisis passed." However, this scene foreshadows the way Jack will rule when he gets the chance, presenting a stark contrast to the British values of freedom and individual rights that Ralph stands for.
Both insist on going after the beast because both are in competition for the leadership position and for the following of the "labor force" of the boys on the island. They recognize that in order to completely secure the title of "leader" they must have the majority of the boys on their side. Ralph wants to put the fear the boys have of the beast to rest--his logical side and leadership style coming out. Jack wants to use this fear to his advantage--his leadership style--and put the entire "tribe" on the defensive with spears, body paint, and and hunter mentality.
Jack renounces the conch shell because it is a symbol of order and civilization...they are quickly moving away from the civilized life from which they came to a more barbaric "dog eat dog" world.