“Not to the purpose” is appropriate, but it is both too broad in scope and too vague. It covers digressions and innocent irrelevancies, which are not invariably instances of bull; furthermore, saying that bull is not to the purpose leaves it uncertain what purpose is meant. The reference in both definitions to “hot air” is more helpful. When we characterize talk as hot air, we mean that what comes out of the speaker’s mouth is only that. It is mere vapor. His speech is empty, without substance or content. His use of language, accordingly, does not contribute to the purpose it purports to serve. No more information is communicated than if the speaker had merely exhaled. There are similarities between hot air and excrement, incidentally, which make hot air seem an especially suitable equivalent for bullshit. Just as hot air is speech that has been emptied of all informative content, so excrement is matter from which everything nutritive has been removed. Excrement may be regarded as the corpse of nourishment, what remains when the vital elements in food have been exhausted. In this respect, excrement is a representation of death which we ourselves produce and which, indeed, we cannot help producing in the very process of maintaining our lives. Perhaps it is for making death so intimate that we find excrement so repulsive. In any event, it cannot serve the purposes of sustenance, any more than hot air can serve those of cummunication.
Now consider these lines from Pound’s Canto LXXIV, which the OED cites in its entry on bullshit as a verb: