Mishnayos 8 and 9 deal with another mixture of blood: two types of blood that mixed together, where the place for applying one is above the red line, and the place for applying the other is below it.
The Places of Application:
The blood of a chatas is applied above the red line (the chut hasikra), on the four corners of the Altar, while the blood of all other offerings is applied below the line. The case before us is a mixture of the blood of a chatas and the blood of an olah, which were all mixed together.
The Opinion of Rabbi Eliezer:
First Application Above: He first applies the blood above the line on the four corners, following the law of a chatas. Seemingly, there is a problem here, as he is applying blood that belongs below the line above it; however, Rabbi Eliezer teaches that there is no explicit prohibition against this, and it is sufficient for him to have explicit intent that he is not applying this blood as the blood of an offering. His intention will be that the blood applied to the Altar is only the blood of the chatas, while the other blood is treated "k'mayim" - like water, which is considered as nothing, and is neither here nor there.
Second Application Below: Afterwards, he makes two applications below the line, on the northeastern and southwestern corners, for the blood whose place is below. These applications fulfill two obligations at once: the obligation to apply the blood of the olah on the four walls of the Altar, and the obligation to pour the leftover blood onto the base. This is because one who applies blood on the wall below the chut hasikra, above the base, is considered as pouring the leftovers onto the base, even though the blood is not applied directly onto the base itself. It turns out that in the second stage, everything is done properly, and there is no need to even treat the blood like water, thus solving the problem.
The Opinion of the Sages:
The Sages disagree: A person has no business applying blood above the red line, and even though there is no explicit prohibition in itself - this is not its place and this is not the way of application. Therefore, they reject the solution of having intent that "it is as if it were water, as if it were not there," because in practice he is applying the blood above, in a place where it does not belong.
This creates a situation of mutual contradiction: One cannot throw the blood mixture above the line because of the blood of the olah, and one cannot throw it below the line because of the blood of the chatas - and therefore, they do not throw it at all. They wait until the sun sets, at which point all the blood becomes invalidated and can no longer be thrown, and its ruling is "yishafeich el ammas hamayim" - it shall be poured into the water channel, the drainage system in the Courtyard of the Temple.
"V'im lo nimlach v'nasan" - And if he did not consult and applied it:
A kohen who did not consult or ask how to act with a mixture of chatas blood and olah blood, but rather applied the blood in its usual order - first above the chut hasikra following the law of chatas blood, and afterwards below - his application is valid. This is because in practice he did what was required of him: he applied the chatas blood in its place and the olah blood in its place. Even though according to the Sages one should not do this initially, after the fact, the applications are effective.
Furthermore: Even if he initially applied the blood only on the four upper corners, even the Sages concede that he may continue and apply it on the two lower corners, because after the fact, the chatas blood also belongs there, and all the bloods are valid. Once the first stage of application is done above the chut hasikra following the law of the chatas, the application for the other offering can go down below, and such is the halachah.
In summary: In a mixture of blood whose place is above the red line with blood whose place is below it, Rabbi Eliezer holds that one must first apply it above following the law of a chatas, with the explicit intent that the other blood is "like water," and afterwards make two applications below which fulfill both the application of the olah blood and the pouring of the leftovers onto the base. The Sages reject this and rule that they do not throw the blood at all, but rather wait for sunset, and the blood becomes invalidated and is poured into the water channel. However, if he did not consult and applied it, the application is valid after the fact, and even according to the Sages, once he has begun with the applications above, he is permitted to complete the applications below.