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Śrīmad-Bhāgavatam
[Canto 10, Ch. 3]
appeared in My original form as Śrī Kṛṣṇa.” Śrīla Jīva Gosvāmī says that the other forms were partial expansions of the Lord’s original form, but because of the intense love developed by Pṛśni and Sutapā, the Lord appeared from Devakī and Vasudeva in His full opulence as Śrī Kṛṣṇa. In this verse the Lord confirms, “I am the same Supreme Personality of Godhead, but I appear in full opulence as Śrī Kṛṣṇa.” This is the purport of the words  tenaiva vapuṣā . When the Lord mentioned the birth of Pṛśnigarbha, He did not say  tenaiva vapuṣā , but He assured Devakī that in the third birth the Supreme Personality of Godhead Kṛṣṇa had appeared, not His partial expansion. Pṛśnigarbha and Vāmana were partial expansions of Kṛṣṇa, but in this third birth Kṛṣṇa Himself appeared. This is the explanation given in Śrī Kṛṣṇa- sandarbha  by Śrīla Jīva Gosvāmī.
TEXT 44
एतद् वां दर्शितं रूपं प्राग्जन्मस्मरणाय मे । नान्यथा मद्भवं ज्ञानं मर्त्यलिङ्गेन जायते ॥४४॥
etad vām darśitam rūpam prāg-janma-smaraṇāya me nānyathā mad-bhavam jñānam martya-liṅgena jāyate
etat —this form of Viṣṇu;  vām —unto both of you;  darśitam —has been shown;  rūpam —My form as the Supreme Personality of Godhead with four hands;  prāk-janma —of My previous appearances;  smaraṇāya —just to remind you;  me —My;  na —not;  anyathā —otherwise;  mat-bhavam —Viṣṇu’s appearance;  jñānam —this transcendental knowledge;  martya-liṅgena —by taking birth like a human child;  jāyate —does arise.
TRANSLATION
I have shown you this form of Viṣṇu just to remind you of My previous births. Otherwise, if I appeared like an ordinary human child, you would not believe that the Supreme Personality of Godhead, Viṣṇu, has indeed appeared.