What motivates Bernard and Helmholtz’s friendship?

Bernard and Helmholtz are very different characters, and they are motivated in this friendship by different things. Though he is an Alpha, Bernard does not have the finer things in life that Helmholtz has, like physical stature, lots of beautiful women, and the respect of his peers. Bernard, “wishes that he could have as many girls as Helmholtz did, and with as little trouble” (p. 72), and he wants to be friends with Helmholtz in hopes that some of Helmholtz’s glory will rub off on him. This is not quite hero-worship on the part of Bernard, but it is not far off.
Helmholtz, on the other hand, is bored with his success and the ease with which he accomplishes any goal. He takes pity on Bernard, in the way an older brother takes pity on a young sibling who is trying to emulate the older brother’s success. Also, Helmholtz sees Bernard’s dissatisfaction with the Director and their world in general, and Helmholtz appreciates this feeling. Bernard expresses his dissatisfaction through continuous comments in defense of Lenina’s sexuality, which he believes should be more highly valued. Helmholtz shares Bernard's disillusionment with the world, going so far as to write a series of rhymes that contradict the sleep-teaching to which everyone is subjected. This brings him into trouble with the Authority (p. 164). In this way, they see each other as mutual outcasts and are each seeking the comfort of a fellow rebel.