The most important successor to Nāgârjuna in the development of this school was Āryadeva 聖提婆 (170–270, also commonly written 提婆) who helped to complete the framework of the Madhyamaka system in his Catuh-śataka 四百論, etc. He was followed a few centuries later by such figures as Buddhapālita (佛護, c.470–540), Bhāviveka (淸辯, c. 490–570) and Candrakīrti (月稱, c. 650), who worked out, each in their own ways various types of proofs for the explanation of and apprehension of the nature of emptiness. Some of the most important basic antagonistic positions within the broad Madhyamaka tradition can be seen in the disagreement between Bhāviveka and Candrakīrti: while Bhāviveka used one form of logical argumentation to establish the truth of emptiness in a positive manner, Candrakīrti denied the validity of the use of logical propositions which ended up affirming any sort of positive position. This basic disagreement would become embedded within the Madhyamaka school to the extent that two distinct philosophical factions developed: the Svātantrika (獨立論證派, which basically followed the position of Bhāviveka and the Prâsaṅgika 歸謬論證派, which supported the opinion of Buddhapālita and Candrakīrti. Some later (eighth century) Mādhyamika thinkers such as Śāntarakṣita 寂護 and Kamalaśīla 蓮華戒 blended the Mādhyamika theories together with that of the Yogâcāra movement, which had risen in popularity during the fifth century. The Three-treatise school 三論宗 which arose in East Asia was based primarily in the basic positions of the schoolʼs earliest thinkers, Nāgârjuna and Āryadeva. (Tib. dBu ma pa'i lugs)

References

the Internet Encyclopedia of Philosophy has a very good article about thirty pages in length, written by Dan Arnold. Some other recommended books include;

Arnold, Dan. 2005. Buddhists, Brahmins, and Belief: Epistemology in South Asian Philosophy of Religion. New York:  Columbia University Press.

Bhattacharya, Kamaleswar. 1990. The Dialectical Method of Nāgârjuna: Vigrahavyāvartanī. Delhi:  Motilal Banarsidass.

Blumenthal, James. 2004. The Ornament of the Middle Way: A Study of the Madhyamaka Thought of Śāntarakṣita. Ithaca, NY:  Snow Lion Publications.

Crosby, Kate, and Andrew Skilton, trans. 1995. The Bodhicaryâvatāra. New York:  Oxford University Press.

Garfield, Jay L., trans. 1995. The Fundamental Wisdom of the Middle Way: Nāgârjunaʼs Mūlamadhyamakakārikā. New York:  Oxford University Press.

Huntington, C. W., with Geshe Namgyal Wangchen. 1989. The Emptiness of Emptiness: An Introduction to Early Indian Mādhyamika. Honolulu:  University of Hawai`i Press.

Murti, T. R. V. 1960. The Central Philosophy of Buddhism: A Study of the Mādhyamika System. London:  George Allen and Unwin. Second edition

Nagao Gadjin. 1991. Mādhyamika and Yogâcāra: A Study of Mahāyāna Philosophies. Albany:  SUNY Press.

Sprung, Mervyn, trans. 1979. Lucid Exposition of the Middle Way: The Essential Chapters from the Prasannapadā of Candrakīrti. London:  Routledge and Kegan Paul.

Walser, Joseph. 2005. Nāgârjuna in Context: Mahāyāna Buddhism and Early Indian Culture. New York:  Columbia University Press.