"Consciousness, then, [is not] chopped up in bits. Such words as 'chain' or 'train' do not describe it fitly as it presents itself in the first instance. It is nothing jointed; it flows. A 'river' or a 'stream' are the metaphors by which it is most naturally described". (James, p239)
The idea (but not the term) had been noted before James, by the philosopher William Hamilton in his lectures of 1838/9. He described thought as "a series of acts all... connected".
A similar idea occurs in early Buddhist writings, and is sometimes called the mindstream.