Early two-author math papers The middle of the twentieth-century featured several famous papers with two authors.  For example, Eilenberg and Mac Lane's papers introducing categories and Eilenberg-MacLane spaces appeared in 1945.  The Feit-Thompson Odd Order Theorem appeared in 1962.  Atiyah and Singer published their index theorem in 1963.
I can't think of any important papers with two or more authors before the Eilenberg-Mac Lane collaboration, which could just be a lacuna in my historical knowledge.  My question is: what are the first math papers with two or more authors?  (A subsidiary question is: why were collaborations so rare before that?)
 A: In differential geometry a fundamental collaborative paper is: 
Ricci, Gregorio, Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications". Mathematische Annalen (Springer) 54 (1–2): 125–201. doi:10.1007/BF01454201.
Its digitalization is freely available here.
Edit: I replaced the link to a pay-walled copy with another to the GDZ repository.
A: As far as the number of joint authors is concerned, this one is hard to beat: Malmsten, C.J., Almgren, T.A., Camitz, G., Danelius, D., Moder, D.H., Selander, E., Grenander, J.M.A., Themptander, S., Trozelli, L.M., Föräldrar, Ä., Ossbahr, G.E., Föräldrar, D.H., Ossbahr, C.O., Lindhagen, C.A., Moder, D.H., Syskon, Ä., Lemke, O.V., Fries, C., Laurenius, L., Leijer, E., Gyllenberg, G., Morfader, M.V., Linderoth, A.: Specimen analyticum, theoremata quædam nova de integralibus definitis, summatione serierum earumque in alias series transformatione exhibens (Eng. trans.: "Some new theorems about the definite integral, summation of the series and their transformation into other series") [Dissertation, in 8 parts]. Upsaliæ, excudebant Regiæ academiæ typographi. Uppsala, Sweden (April—June 1842) 
Edit (December 20, 2016): t turns out the information in Blagouchine's citation is imprecise and the number of Malmsten's co-authors has to be reduced. More detailed information which I received from Lennart Börjeson (thanks!) indicates first of all that Moder, Föräldrar, Syskon and Morfader are not last names but rather parts of respective dedications (translation: mother, parents, siblings and maternal grandfather). Indeed, in the scanned copy of the paper available here, goo.gl/bZIWZZ one such dedication can be seen on p. 2. Another point: the remaining names are those of students and the parts of the paper are probably their theses. This makes the co-authorship questionable, since before 1852 in Sweden it was common that the professor wrote the thesis bearing the student's name rather than the student himself. Still, the names appear jointly in print (only some of them can be seen here, but the paper version from which the scan was made might have been a part of a bigger series of publications).
A: Obvious answer to subsidiary question: communication was vastly more difficult then, and travel similarly. For that matter, in the U.S., prior to the advent of the internet, it was crazily slow to communicate with anyone far away (long-distance phone calls cost an absurd amount, too much to "discuss" anything, even if paid-for by an NSF grant... and physical mail would take a week each direction, due to the additional slow-down of campus mail interface).
Further, "in the old days", people up for tenure or promotion would vastly have preferred solo papers, because there was more presumption (I guess due to the predominance of solo papers...) that a joint paper was surely much less a product of the junior author... or something.
Even the now-mundane issues of revision would have been grossly complicated in those days: no TeX, so one usually gave a hand-written ms. to a technical-typing-secretary, who would have a bit of technology ("IBM selectric"?) unavailable to most mortals, and produce a presentable mostly-typed document. But introduce errors. To photocopy and physical-mail to co-author far away would introduce an extra two-week delay...
The practical difficulties certainly explain the relative ease of contemporary collaboration, and, thus, the relative plausibility, etc.
A: Joint mathematical papers were pioneered by Polish School after WWI. Offhand, let me mention popular Knaster-Kuratowski-Mazurkiewicz paper on the fixed point theorem or Borsuk-Ulam paper on antipodes or Banach-Tarski paradox. There were many more.
A: *

*Gergonne’s Annales contains many nominally “multi-authored” articles. Most are just collations of independent parts (e.g. “solutions of problems”) by the editor, but the paper

Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatre conditions données
Par MM. BRIANCHON , capitaine d’artillerie , professeur
  de mathématiques à l’école d’artillerie de la garde
  royale , et PONCELET , capitaine du génie , employé à Metz.
Annales de mathématiques pures et appliquées XI, nº VII (1821), pp. 205-220

 seems to be both joint work and older than the (imho) earliest genuinely collaborative papers yet mentioned on this page (viz., Sturm and Liouville 1836-1837).

*Earlier (but could be regarded as teaching rather than original research): Monge & Hachette, Application d’algèbre à la géométrie; Hachette & Poisson, Addition au mémoire précédent, both in Journal de l’école polytechnique IV, nº XI, (1802).

*Earlier (but maybe subject to the same objections as Theoria motuum lunae): Théorie de la lune (1770 prize essay, labeled “par MM. Léonard Euler & J. A. son fils, conjointement”; Johann Albrecht Euler has 31 papers of his own.)

*Earlier (and original, but not a “paper”): Le Seur & Jacquier, Elémens du calcul intégral (1768).
A: The famous paper of Dedekind and Weber:
R. Dedekind, H. Weber: Theorie der algebraischen Functionen einer Veränderlichen, J.
Reine Angew. Math 92 (1882) 181-290.
is the first place where the points of a Riemann surface are described in terms of ideals of the ring of functions. To put this into context, Dedekind had only invented the notion of ideal a few years earlier. They also give an algebraic proof of the Riemann-Roch theorem.
I think the analogy between function fields and number fields started here.
A: The Brill--Noether paper appeared in 1874: "Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie". Math. Annalen 7, 269–316.
A: Abraham Ecchellensis (Ibrahim Al-Haqilani) and Giovanni Alfonso Borelli published in 1661 a Latin translation of the 5th, 6th and 7th books of the Conics by Apollonius of Perga. Both Ecchellensis's and Borelli's names are on the title page, a highly unusual feature at the time. 
For more details, see the paper Authorship and Teamwork Around the Cimento Academy by Domenico Bertoloni Meli, available here. 
A: Sturm and Liouville published joint papers in 1836-1837.
A: There was a known paper by Alexandrov-Urysohn on compact spaces (y. 1929) -- they called them bicompact spaces in the Soviet Union for a long time.
A: Though it was initially likely more intended (and perceived) as physics, the paper which introduced what has since then become known as the Korteweg-De Vries equation [Korteweg, D. J.; De Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves., Phil. Mag. (5) XXXIX, 422-443 (1895). JFM 26.0881.02] may be counted here due to its lasting influence on the analysis of PDE.
A: Felix Klein and Sophus Lie published at least a couple papers together:


*

*Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen. Mathematische Annalen 4 (1871): 50-84. http://eudml.org/doc/156514.

*Ueber die Haupttangentencurven der Kummer'schen Fläche vierten Grades mit 16 Knotenpunkten. Mathematische Annalen 23 (1884): 579-586. https://doi.org/10.1007/BF01446604
(These are the joint articles listed at https://de.wikisource.org/wiki/Sophus_Lie, I do not know if there are others.)
A: Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen by
N.H. Abel, Th. Clausen, J. Steiner, published in 1827, Journal für die reine und angewandte Mathematik. Volume 1827, Issue 2, Pages 286–292. I think every paper by Abel must be important.  
Also:
Nouvelles formules analogues aux séries de Taylor et de Maclaurin. by B. Clapeyron
G. Lamé,  in Journal für die reine und angewandte Mathematik. Volume 1830, Issue 6, Pages 40–44. 
I do not know who was Clapeyron, but Lamé is well known. These two papers may be the first collaborative journal papers because Crelle was the first math journal (or one of the first journals). But this site: http://www.daviddarling.info/encyclopedia/B/Bernoulli.html talks about a joint paper in astronomy produced by Bernoulli family. At that time (~1730) astronomy and mathematics were not that far apart. 
A: Here is one from the eighteenth century. There must be others.
Theoria motuum lunae, nova methodo pertractata una cum tabulis astronomicis, unde ad quodvis tempus loca lunae expedite computari possunt incredibili studio atque indefesso labore trium academicorum: Johannis Alberti Euler, Wolffgangi Ludovici Krafft, Johannis Andreae Lexell. Opus dirigente Leonhardo Eulero acad. scient. Borussicae directore vicennali et socio acad. Petrop. Parisin. et Lond. Petropoli, typis academiae imperialis scientiarum. 1772.
A: The first explicit construction $E_8$ lattice  (positive definite even unimodular quadratic form in $8$ variables) was given by A. Korkin and G. Zolotarev in 1873, see "Sur les formes quadratiques". Mathematische Annalen. 6: 366–389.
A: Very important joint work one of whose authors is a somewhat well-known mathematician, but not a mathematics paper was the construction of the first electromagnetic telegraph by C. F. Gauss and W. E. Weber (1833), and then:
Gauss, Carl Friedrich; Weber, Wilhelm Eduard (1840). Atlas Des Erdmagnetismus: Nach Den Elementen Der Theorie Entworfen. Leipzig: Weidmann'sche Buchhandlung.
Don't know if that counts, but I don't know of any joint work of Gauss in pure mathematics...
A: The work introducing the Phragmén–Lindelöf principle should likely also be mentioned here:
Phragmén, E.; Lindelöf, E., Sur une extension d’un principe classique de l’analyse et sur quelques propriétés de fonctions monogènes dans le voisinage d’un point singulier., Acta Math. 31, 381-406 (1908). Zbl 39.0465.01.
A: The Lotka-Sharpe population model was introduced in their joint 1911 paper Sharpe, F. R.; Lotka, A. J., A problem in age-distribution., Phil. Mag. (6) 21, 435-437 (1911). JFM 42.1030.02. and has been much in use in demography since then.
A: The classification of finite groups with only abelian subgroups by _G. A. Miller and H. C.Moreno Non-Abelian groups in which every subgroup is Abelian., American M. S. Trans. 4, 398-404 (1903). ZBL 34.0173.01., which is still frequently used, might also be worth mentioning.
A: Another example worth to be mentioned might be the Bernstein-Doetsch theorem on convex functions [Bernstein, Felix; Doetsch, Gustav, On the theory of convex functions, Math. Ann. 76, 514-526 (1915). JFM 45.0627.02].
A: The Liénard–Chipart stability criterion [Liénard; Chipart, Sur le signe de la partie réelle des racines d’une équation algébrique., Journ. de Math. (6) 10, 291-346 (1914). JFM 45.1226.03.] is still used in control theory.
