Ānupubbī-kathā: 'gradual instruction', progressive teaching; given by the Buddha when it was necessary to first prepare the listener's mind before speaking to him on the advanced teaching of the Four Noble Truths. The stock passage e.g. D. 3; D 14; M. 56 runs as follows:
Then the Blessed One gave him a gradual instruction - that is to say, he spoke on generosity dāna, on morality sīla and on the heavens sagga, he explained the danger, the loss and the depravity of sensual pleasures, and the advantage of renunciation. When the Blessed One perceived that the listener's mind was prepared, pliant, free from obstacles, elevated and lucid; then he explained to him that exalted teaching particular to the Buddhas Buddhānam sāmukkamsikā, that is: Suffering, its cause, its ceasing, and the path.
Maha Thera Nyanatiloka. Manual of Buddhist Terms and Doctrines, Buddhist Publication Society, first edition 1952.