Hare Krsna,
After hearing about the multiple incarnations of the Supreme Lord, I could see a simple pattern. It is my own realization; I could be wrong. The appearance of the Supreme Lord almost always has two aspects – internal reason and external reason. In the appearance of Lord Sri Caitanya Mahaprabhu, the internal reasons are to understand the glory of Radharani’s love, to experience the wonderful qualities in Him that She alone relishes through Her love, and to taste the happiness She feels when She realizes the sweetness of His (Krsna’s) love. (CC Adi 4.230). The external reason for Mahaprabhu’s appearance is His desire to distribute pure love of Godhead (Krsna prema) through nama-sankirtana (congregational chanting of the Holy Name).
Similarly in Lord Krsna’s appearance, there are multiple reasons. One of the internal reasons I could think of is Krsna’s desire to teach the living entities how to perform spontaneous devotional service.
“Mother sometimes binds Me as her son. She nourishes and protects Me, thinking Me utterly helpless. My friends climb on My shoulders in pure friendship, saying, ‘What kind of big man are You? You and I are equal. If My beloved consort reproaches Me in a sulky mood, that steals My mind from the reverent hymns of the Vedas.Taking these pure devotees with Me, I shall descend and sport in various wonderful ways, unknown even in Vaikuṇṭha. I shall broadcast such pastimes by which even I am amazed.” (CC Adi 4.24-28)
And, of course, the external reason of Krsna’s appearance is mentioned in BG 4.7-9:
“Whenever and wherever there is a decline in religious practice, O descendant of Bharata, and a predominant rise of irreligion – at that time I descend Myself. To deliver the pious and to annihilate the miscreants, as well as to reestablish the principles of religion, I Myself appear, millennium after millennium. One who knows the transcendental nature of My appearance and activities does not, upon leaving the body, take his birth again in this material world, but attains My eternal abode, O Arjuna.”
This means that for the Lord’s advent, the internal reason(s) and the social conditions must meet. Regarding the appearance of Lord Kapiladeva, the Supreme Lord Himself announced that He will appear through the conjugal union of Kardama Muni and Mother Devahuti.
“O great sage, I shall manifest My own plenary portion through your wife, Devahūti, along with your nine daughters, and I shall instruct her in the system of philosophy that deals with the ultimate principles or categories.” (SB 3.21.32)
The social condition (external reason) I could think of is the protection of Mother Devahuti. So far we have studied that Kardama Muni desired to leave home after giving birth to nine daughters.
“The powerful Kardama Muni was the knower of everyone’s heart, and he could grant whatever one desired. Knowing the spiritual soul, he regarded her as half of his body. Dividing himself into nine forms, he impregnated Devahūti with nine discharges of semen. Immediately afterward, on the same day, Devahūti gave birth to nine female children, all charming in every limb and fragrant with the scent of the red lotus flower.” (SB 3.23.47-48)
We now know that Kardami Muni is about to leave home and Devahuti for spiritual pursuits. In due course of time, the nine daughters will also get married to their respective husbands and will leave home. Who will give protection to Devahuti after they leave home? In childhood a woman must be under the protection of the parents, in youth she must be under the protection of the husband, and in old age she must be under the protection of the grown children. In Dehavuti’s case, she will be left with no one after her husband and daughters leave home. Thus there is a need for a son to appear from Kardama Muni and Devahuti who will give protection and care to Mother Devahuti. The wonderful thing is that that son will be none other than the Supreme Lord Himself (Lord Kapiladeva).
Another aspect to consider is that while the Supreme Lord desires to increase the population of the material world through the agency of Lord Brahma and His sons/expansions, the Supreme Lord also needs to establish transcendental knowledge. Otherwise, what is the use of increasing population but not teaching them bhakti and devotional service? Through Lord Kapiladeva, the Lord gives protection to Mother Devahuti and also propagates pure devotional service by instructing her.
All these factors are prelude to the appearance of Lord Kapiladeva.
All glories to Srila Prabhupada!