Shulchan Aruch Chapter 246:
The Laws of Lending and Renting to a Gentile on the Sabbath (5 Paragraphs)
Note:  The REMA is in brackets [ ]

1.  It is permitted to lend or rent one's vessels to a Gentile even though he will do work through them on the Sabbath.  The reason is that we were not commanded to have our vessels also rest on the Sabbath.  There are those who hold that it is forbidden if one's vessels have work done through them on the Sabbath; for example plowing.  So it would be prohibited to rent vessels to a Gentile on Friday.  On Thursday it would be permitted to rent one's vessels to a Gentile only on the condition that he does not take the wages of the Sabbath (work) but loans them through a loan.  For example, one rents it to the Gentile for a month or for a week.  To lend vessels, it is permitted even on Friday.  [We hold like the last opinion and it is permitted to lend a vessel to a Gentile on Friday.  Even though the gentile makes a deal to borrow and return the object we do not say that it is like a renter.]

2.  It is forbidden to lend any vessel to a Gentile on the Sabbath.  Even on Friday close to sunset it is forbidden to lend any vessel that will be unable to be brought out of the home before sunset.  This is due to the fear that passers by who witness this will think that the Jew instructed the Gentile to bring the vessels out. 

3.  It is forbidden to lend or to rent out one's animal to a Gentile in order that work would be done on the Sabbath since we were commanded regarding the resting of our animals.  [One may rent or lend one's animal to a Gentile on the condition that it is returned before Shabbat.  It would not help if the Gentile made a stipulation that the animals were to rest on Shabbat since he is not trusted on this.]  If one would rent or lend to a Gentile on the condition that it be returned before the Sabbath and the Gentile kept the animal on Shabbat he should make the animal hefker [1] by himself before the onset of the Sabbath or he should say "my animal becomes the property of the Gentile" in order to save himself from a Biblical transgression.  [If he wishes, he may make it hefker before three people like other acts of hefker.  Even here, another person is not able to take possession of his animal since his intention was only to remove from himself the transgression.  The above only applies on Shabbat.  However, with regard to Yom Tov (festivals) one is not commanded to have his animals rest on these holidays.]

4. A  Jew who rents oxen to a Gentile in order to plow a field or to sow there are those who permit it in a special situation where the Gentile accepted upon himself the seven Noahide laws including kidnapping, murder, stealing, believing in G-d, immorality, eating from a live animal, and civil laws.  Others say that since the Gentile is unable to sell the animal at his will it is still considered the Jew's animal. [2]  [See below in the next paragraph]

5.  If a Jew and a Gentile are partners in owning an animal then it is permitted for the Gentile to work the animal on Shabbat if they made an initial agreement when they purchased the animal that the Gentile is to work the animal on Saturday and the Jew is to work the animal on a weekday.  If they did not make this initial agreement, then it is prohibited even if they made the idential agreement at a later date.  If the Jew loans the animal to a Gentile with a complete loan such that the Gentile has permission to do whatever he wishes with the animal without obtaining permission from the Jew, and charge the value to the Gentile such that the oxen are mortgaged property to the Gentile, then it is permitted.  There are those who permit this even if the Gentile did not have permission to take out the animal since he charges the value to the Gentile and later the Gentile returns the animal.  In effect a pledge is made [he means to say that the Jew stated "you will only have to pay in the following situation [3]"] to the Jew or yehar'hanam [meaning collateral which is called raan in Arabic].  This only applies if he did not tell him "from now"[4].  There are those who permit the above case only where he warned the Gentile to not do work on the Sabbath.  If he transgressed and did work on the Sabbath, then the responsibility will be on the Gentile's, even from accidental transgressions if he wrote down this agreement  since now if he comes and does work on Shabbat it is not a Jewish animal since the Gentile acquired it to be liable for inadvertent transgressions. [All of these situations described above as permissible, such is the law and one may do whatever of them one wishes to do.  Even if the animal completely belonged to the Jew, the law is that it is as if it was a joint partnership with the Gentile solely hat it becomes publicized that it becomes a permissible way.] [5]
[1] Ownerless
[2]  The Mishna Brura offers the following explanation:  The Gentiles of that time period were not trustworthy and would frequently steal from Jews.  It was common for the Gentiles to rent an animal from a Jew and then sell the animal to someone else.  For this reason, when a Jew rented an animal to the Gentile, he was worried about it being stolen so he did not give up any of his ownership rights to the animal for the period of the lease.  So according to Jewish law the animal is still considered the property of the Jew and in the Jew's possession, even if a Gentile rented it and it is in the Gentile's possession.  In the special case where a Gentile fears G-d and accepts upon himself not to steal, the Jew is not afraid that the Gentile will sell his animal and it becomes like it is owned by the Gentile and in the Gentile's property. 
[3] This is a very confusing case.  I think he is saying that the Jew made a deal with the Gentile that the animal was given as a loan with collateral given to the Jew that is returned when the animal is returned.
[4] I am at a loss in explaining  what he is trying to say. 
[5] These last few sentences are confusing and I do not fully understand them.  They seem to contradict paragraph 3 and 4 which state that one may not lend or rent one's animal to a Gentile to be used on the Sabbath.  But then again, it seems this is talking about a case where he did a higher form of lending called "a complete loan". 
Translated by Jay Dinovitser DO 12/11
Free for personal use