Avatar, Avatara (Sanskrit) [from ava down + the verbal root tri to cross over, pass] That which passes down or descends; the passing down of a celestial energy or an individualized complex of celestial energies -- a celestial being -- in order to overshadow and illuminate a human being who, at the time of such connection of divinity with matter, possesses no human soul karmically destined to be the inner master of the body thus born. "Hence an Avatara is one who has a combination of three elements in his being: an inspiring divinity; a highly evolved intermediate nature or soul, which is loaned to him and is the channel of that inspiring divinity; and a pure, clean, physical body" (OG 16). 
Sankaracharya, Krishna, Lao-tzu, and Jesus were avataras in differing degrees, of somewhat differing structure. There was a divine ray which came down at the cyclic time of each of these incarnations, and the connecting link or the flame of mind was provided in each case by a member of the Hierarchy of Compassion. Krishna says, "I incarnate in period after period in order to destroy wickedness and reestablish righteousness" (BG ch 4, sl 8). Krishna here represents the Logos or logoic ray which "on our plane would be utterly helpless, inactive, and have no possible means of communication with us and our sphere, because that logoic ray lacks an intermediate and fully conscious vehicle or carrier, i.e., it lacks the intermediate or highly ethereal mechanism, the spiritual-human in us, which in ordinary man is but slightly active. An avatara takes place when a direct ray from the Logos enters into, fully inspires, and illuminates, a human being, through the intermediary of a bodhisattva who has incarnated in that human being, thereby supplying the fit, ready, and fully conscious intermediate vehicle or carrier" (Fund 276). 
Blavatsky says that "rebirths may be divided into three classes: the divine incarnations called Avataras; those of Adepts who give up Nirvana for the sake of helping on humanity -- the Nirmanakayas; and the natural succession of rebirths for all -- the common law. The Avatara . . . is a descent of the manifested Deity -- whether under the specific name of Siva, Vishnu, or Adi-Buddha -- into an illusive form of individuality, an appearance which to men on this illusive plane is objective, but it is not so in sober fact. That illusive form having neither past nor future, because it had neither previous incarnation nor will have subsequent rebirths, has naught to do with Karma, which has therefore no hold on it" (BCW 14:373-4). 
Vishnu as the supporter of life is the source of one line of avataras so often spoken of in Hindu legends. These ten avataras of Vishnu are: 1) Matsya the fish;
2) Kurma the tortoise; 
3) Varaha the boar; 
4) Narasimha the man-lion (last of animal stage);
5) Vamana the dwarf (first step toward the human form);
6) Parasu-Rama, Rama with the axe (a hero);
7) Rama-chandra, the hero of the Ramayana;
8) Krishna, son of Devaki;
9) Gautama Buddha; and
10) Kalki, the avatara who is to appear at the end of the kali yuga mounted on a white horse, inaugurating a new reign of righteousness on earth. A horse has from immemorial time been a symbol of the spiritual as well as vital energies of the inner solar orb. Hence, when the next avatara is said to come riding a white horse, the meaning is that he comes infilled with the solar light or splendor -- an avatara or manifestation of a spiritual and intellectual solar energy which will carry all before it on earth. 
Brahmanical esotericism never taught that divinity descended into the animals as given in the legends. These names of different animals and men, like all zoological mythology, were chosen because of certain characteristic attributes. They actually represent ten degrees of advancing knowledge and growth in understanding -- ten degrees in the esoteric cycle -- as well as different evolutionary stages through which monads break through the lower spheres in order to express themselves on higher rungs of the evolutionary ladder of life. These names also represent the technical names given to neophytes in esoteric schools. The lowest chela was called a fish, the chela who had taken the second degree successfully was called a tortoise, and so forth, till the highest of all was called an incarnation of the sun -- a white horse in Hindu legend. 
These avataric descents do not appertain solely to a race, root-race, globe, chain, or solar system, because nature repeats itself by analogy, and the same line of enlarging understanding of evolutionary development takes place in all the spheres mutatis mutandis. Thus these avataric descents can be ascribed to the solar system, the planetary chain as a whole, a globe, a root-race, and even to a subrace.