כי ימצא חלל באדמה, "If a slain person is found lying on the earth, etc.⁠" According to the plain meaning of the text the legislation about breaking the neck of the עגלה ערופה, the calf the Torah commands to be killed publicly and to be thrown into a virgin valley, is supposed to be a great trick designed to reveal the identity of the murderer. Maimonides in Moreh Nevuchim 3,40 writes as follows: "the people killing this calf are from the town nearest where the murdered person has been found. In most instances the murderer is a local resident. The elders of that town testify that they had not been negligent in maintaining all the services which are part of a civilised town. It is hoped that the publicity which this procedure attracts will lead to information disclosing the identity of the culprit.⁠"
The sages are already on record that even an unsubstantiated statement by a lowly slave-woman claiming that a certain person is the murderer is sufficient to halt these proceedings (Jerusalem Talmud Sotah 9,1). If the identity of the murderer was known but the townspeople were part of a conspiracy of silence, and they have the audacity to testify in front of their Creator that they did not know the identity of the murderer, this is a great sin and anyone (from outside) who acquires knowledge is called upon to make public what he knows, so that the guilty party will be dealt with either by a court of law, the king, or the relative who is the blood-avenger. The whole incident assumes additional significance in that the site where the calf was killed will forever have to remain virgin earth so that people who observe that this earth is not being used for anything will make inquiries about this. Thus far Maimonides.
The expression וערפו means that they are to remove the neck of this calf (Ibn Ezra). The reason why all of this has to take place in the town nearest the slain person is that in our estimation the murderer comes from that city, or that people of that town had been guilty of a similar crime; otherwise such a disaster would not have happened near their town.

Some explain the words נחל איתן, as a place of דשן ושמן, a very fertile valley. Now that this tragedy has occurred, the Torah forbids us to again make use of that land. Our sages in Sotah 15 also understand the words אשר לא יעבד בו ולא יזרע as referring to the future; after the calf has been killed there, this site is to be treated as if out of bounds completely.
The Mishnah in Sotah 45 says that although this site is forbidden to be sown, one is allowed to comb flax on that site or to chisel stones there seeing that these activities are not performed with the actual soil. According to this latter explanation every one of us must be careful to maintain roads in such a condition including security patrols, so that murder cannot occur and the murderer escape. Once a property owner knows that if such a murder has occurred nearby they may all forfeit use of their property as a result he will cooperate in supervising his town and what is around it (Based on the father of R' David Kimchi in the Sefer Hashoroshim).

A Midrashic approach to the subject of עגלה ערופה, based on Sotah 46: God said: "let the yearling calf whose neck is broken serve as a symbol and have her neck broken in a place which will not produce fruit (any produce) in order to atone for the slain person whose life has been cut short so that he will no longer be able to produce "fruit.⁠"
Other commentators dealing with the legislation surrounding the עגלה ערופה claim that the reason the calf in question must be one that has never born a yoke or performed labor for man, and that it has to be brought to a valley which has no traces of ever having been worked by man is so the calf will not recognize the site. There is to be no paved path through that valley so that the calf is not tempted to leave that area. The priests would hit it on its neck in order to encourage it to escape that valley and show us the way to the murderer. This is the meaning of the words (verse 4) וערפו שם את העגלה בנחל. Prior to that the elders of the town washed their hands over the calf, saying: "our hands have not spilled this blood.⁠" The priests would then respond: "grant atonement for Your people Israel.⁠" The expression כפר in this instance includes a request for the facts to come to light, for the murderer to be found. If the Israelites of that generation were generally Torah-observant, the calf would walk to the house of the murderer, stop, and the priests would proceed to kill the calf at that spot. This is the meaning of the words: "and the blood shall be atoned for them.⁠"
If the people of that generation were generally not Torah-observant, and therefore entitled to identify and deal with the murderer, and as a result the calf would not seek out a specific house before coming to a halt, the priests would hit the neck of the calf with an axe, severing the neck. This is the meaning of ונכפר להם הדם, "the blood (guilt) will be atoned for them.⁠" God would have to do the atoning then.

A rational approach to this legislation: the calf which is being killed, עגלה ערופה, is a similar phenomenon to the שעיר לעזאזל, the scape-goat, which is killed after it (symbolically) carries the sins of the entire Jewish people (Leviticus 16,10). Both wind up in a totally barren environment. Something similar happens to one of the birds offered by the person who has been declared cured of the dreaded skin disease tzoraat. It is released into the air after its partner has been slaughtered and the surviving bird's blood has been dipped in the blood of the bird which has been slaughtered (Leviticus 14,51-53). The surviving bird is perceived as carrying the guilt of the person for whom it atones into certain nether regions inhabited by demons and the like. All of the three phenomena we listed have as their purpose to blunt possible accusations by spiritually negative forces which abound in the world and which look for any excuse to assume the role of prosecuting attorney against the Jewish people.
From a technical point of view it would have been enough for the Torah to write that the valley in question should be one which has neither been worked nor sown, i.e. לא יעבד ולא יזרע. Why did the Torah add the extra word בו (verse 4)? The Torah wanted to make clear that not only the valley but also the calf itself must not have performed any kind of work. By being so totally undomesticated, the animal is perceived as (still) belonging to the domain of the forces appointed by God to rule in the desert and similar inhospitable places (as we explained on other occasions). When the calf's neck is broken then it occurs in the domain of these demonic forces. The very word איתן implies that the valley is very harsh, hard, tough.
The reason that the Torah spelled the word שפכה with the letter ה at the end, instead of the letter ו which we would have expected, may be an allusion to the fact that if one kills a person one deprives him of his five senses. By doing so, the murderer has destroyed an entire world. The world had been created with the last letter ה of the tetragrammaton (Bereshit Rabbah 12,9). The reason for this somewhat exaggerated sounding description of the murder of a single individual is the fact that God created only a single human being at the beginning. If he had been killed prematurely, the human race would have been exterminated as the result of such a single act of killing.
The reason for the words כפר לעמך ישראל אשר פדית ה', "provide atonement for Your people Israel, whom You have redeemed o Lord" (verse 8), is simple. The procedure of killing this calf resulted in our providing these demonic forces with a delight. This enhances the power of these forces to act as advocates for the prosecution against Israel. As a result, we have to ask forgiveness, כפרה, from Hashem for having been forced to make use of this legislation. Similar considerations are valid in connection with the שעיר לעזאזל. In Leviticus 16,33, at the conclusion of the whole procedure, the Torah writes: "He shall bring atonement upon the Holy of Holies, and upon the Tent of meeting and the Altar; he will bring atonement for the priests and all the people.⁠" In other words, after the whole Yom Kippur service has been dealt with the Torah had to write these words to show that having somehow caused satisfaction to the Azzazel, the people and all concerned are in need of atonement for this.

There is yet another aspect embodied in the procedures of the עגלה ערופה, and that is that the procedures teach us something about the nature of a murdered person. The procedures leading to atonement of this crime comprise three elements. 1) the breaking of the calf's neck; 2) the location where this has to take place, i.e. a totally barren area, one which has never been useful. 3) the recital of a confession by the priests: "our hands did not spill this blood. Please atone for Your people of Israel.⁠"
Every intelligent person realizes that the murderer achieved his objective only by means of separating, detaching his victim from three "souls,⁠" life-forces. He separated him from the animalistic soul, from the soul which enables vegetable matter to grow, and, finally, he separated him from the נפש השכלית, the abstract life-force we call "soul.⁠" We know that the first soul which departs from the slain person is the animalistic soul. This is why the Torah selected a calf for this procedure, seeing that it (the mother) symbolizes animalism. Killing it removes that "soul.⁠" The location, i.e. one that does not produce any growth, symbolizes the "vegetable" soul, i.e. its absence. Finally, the confession and request for atonement symbolise the loss of the abstract soul of the slain person.
I have heard that the carcass of the calf generates worms which will attack the murderer wherever he is. This is a mystery of nature related to the barren valley, the domain of the demons we mentioned.

A Kabbalistic approach: the whole procedure of "sacrificing" the עגלה ערופה to the attribute of Justice in its most severe manifestation aims at preventing this attribute which is poised to strike the community in whose proximity the murder occurred from causing harm. The reason the neck of the animal was chosen as the point of death is that it represents the "hardest,⁠" most unbending part of the personality. Seeing that at this time God, as it were, turns His back (neck) on us, this calf does not have its throat slit as do the animals which are ritually slaughtered. The reason why this whole legislation is described in such detail is that the murder has occurred in Eretz Yisrael, on holy soil. The reason the Torah added the word בקר to the word עגלת בקר (a calf is always a junior בקר) is that this animal symbolises the חיה with the face of an ox in the prophet's vision in Ezekiel 1,10. Sometimes the חיה described there as on the "left,⁠" does appear on the right, i.e. represents the attribute of Mercy.
Although the calf must not have been worked, borne a yoke, it may impose such a yoke, burden on others (symbolically speaking).
As to the words והורדו, "they shall bring it (the calf) down" (verse 4), this means that the elders shall drag the calf down to that valley. The words וערפו שם, reminds us of Genesis 49,24 משם רועה אבן ישראל, [if I understand correctly a reference to Joseph dragging down his family to Egypt, Joseph representing the "ox,⁠" בכור שורו. Ed.]. The words את העגלה בנחל, mean that the calf is to be submerged in the waters of the brook of that valley. These waters are called נחלי אל, a reference to the emanations. The reason the Torah refers to the murdered person as חלל, usually understood as an empty space, is that he has been deprived of his souls, thus effectively making him "empty,⁠" חלול.
The word is also related to חולין, something profane, devoid of sanctity. The procedure is designed to eliminate the spirit of impurity outside the town in the field. This is why the Torah emphasizes נופל בשדה, "fallen in the field" (verse 1). The word חלל is an allusion to orbit, i.e. the revolving planets in their orbits; the slain person is perceived as having been on earth before in a different body. This may be the reason the Torah speaks of חלל באדמה instead of חלל בארץ.
This whole procedure was performed by the priests, the Levites, seeing it is their function to settle strife and plague (verse 5). The words ריב and נגע refer to strife both in the terrestrial regions and in the celestial regions. The waters with which the elders wash their hands (verse 6) represent the waters of the attribute חסד which have their origin in a domain above that of the עגלה. The washing of the hands is to ensure that the elders will not be harmed by the attribute of Justice.