But this was only the commencement of the service. Women might bring their sacrifices into the Great Court; but they might not perform the second rite—that of laying on of hands. This meant transmission and delegation, and implied representation; so that it really pointed to the substitution of the sacrifice for the sacrificer. Hence it was always accompanied by confession of sin and prayer. It was thus done. The sacrifice was so turned that the person confessing looked towards the west, while he laid his hands between the horns of the sacrifice, and if the sacrifice was brought by more than one, each had to lay on his hands. It is not quite a settled point whether one or both hands were laid on; but all are agreed that it was to be done ‘with one’s whole force’—as it were, to lay one’s whole weight upon the substitute.2 If a person under vow had died, his heir-at-law took his place. The only public sacrifices in which hands were laid on were those for sins of public ignorance, when the ‘elders’ acted as representing the people—to which some Rabbinical authorities add public sin-offerings in general,4—and the scapegoat on the Day of Atonement, on which the high-priest laid his hands. In all private sacrifices, except firstlings, tithes, and the Paschal lamb, hands were laid on, and, while doing so, the following prayer was repeated: ‘I entreat, O Jehovah: I have sinned, I have done perversely, I have rebelled, I have committed (naming the sin, trespass, or, in case of a burnt-offering, the breach of positive or negative command); but I return in repentance, and let this be for my atonement (covering).’ According to Maimonides, in peace-offerings a record of God’s praise, rather than a confession of sins, was spoken. But, as the principle prevailed that frequent confession even without sacrifice was meritorious, another formula is also recorded, in which the allusion to sacrifices is omitted.
Closely connected with this was ‘the lifting and waving’ of certain sacrifices. The priest put his hands under those of the offerer, and moved the sacrifice upwards and downwards, right and left; according to Abarbanel also ‘forwards and backwards.’ The lamb of the leper’s trespass-offering was waved before it was slain; private peace-offerings, only after they had been slain; while in public peace-offerings, the practice varied.

Edersheim, A. (2003). The Temple, its ministry and services as they were at the time of Jesus Christ. (pp. 113–115). Bellingham, WA: Logos Bible Software.