Zevachim 81 - Inside and Outside Bloods
If bloods that are applied inside the Sanctuary (like those of Yom Kippur sacrifices) mixed with bloods applied outside on the Altar (most regular sacrifices), then the resulting mixture cannot be applied either outside or inside and has to be poured out into the canal. Note that the author of this ruling does not agree with the principle we saw earlier that blood applied to the wrong place can be viewed as water.
If, however, the Kohen went ahead and applied the mixture outside, and then inside, it is valid. The only potential problem could have been when outside blood is brought inside, and this did not happen here. If, however, the Kohen applied it first inside and then outside, then Rabbi Akiba declares the outside blood invalid - for the very reason we just mentioned, that of outside blood being first brought inside. The Sages declare the blood valid in all cases except of a sin-offering.
The underlying reason for their disagreement is this: the Torah prohibited the blood on a sin-offering from entering the Sanctuary. According to Rabbi Akiba, this applies to all sacrifices, but according to the Sages, it does not.
Art: Maurycy Gottlieb - Jews praying in the synagogue on Yom Kippur