Is there a Planet X?
Introduction
In 1846 the Planet Neptune was discovered. The existence of the planet explained the residuals in the orbit of Uranus. Its residuals shrank to only a fraction of an arcsecond, which is comparable in size to many known possible sources of systematic errors in observations. However, astronomers still kept searching for another planet, which led to the discovery of Pluto by Tombaugh in 1930. To completely remove the residuals in the orbit of Uranus and the residuals in the orbit of Neptune as well, Pluto's mass would have to be between 0.5 Me (Earth mass) and 5 Me, but its mass is only 0.002 Me, far too small to have any detectable effect on the orbits of Uranus and Neptune. Therefore the theoretical search for a tenth planet, Planet X, began, still only based on the small anomalies in the orbits of Uranus and Neptune. The place to look for was the outer region of our solar system, beyond Pluto. According to the Titius-Bode law ( R(i)=0.4 + 0.15*2^i , where R is the semimajor axis) the semimajor axis should be 77.2 AU, taking i=9. However, as is explained in the subquestion "is our solar system typical when compared to others", this law already breaks down for Pluto and Neptune, so this value doesn't mean much.
The search for Planet X
A number of articles have been written about the possible position of Planet X and the effect the planet would have on the orbits of the other planets of our solar system.
In may of 1988 Dr. A.A. Jackson and Dr. R.M. Killen wrote an article in which they integrated four test orbits of a trans-Plutonian planet foward for four million years. They tried to determine the effects of such a body on the stability of the Neptune-Pluto 3:2 resonance, which means that when Neptune makes three orbits, Pluto makes only two. They found that such a planet with masses of 0.1 Me and 1.0 Me at a distance to the Sun of 48.3 AU and 75.5 AU respectively, do not have any effect on the resonace of Neptune and Pluto. Planets of mass 5 Me with semimajor axes of 52.5 AU and 62.5 AU however, do have an effect. It is therefore unlikely according to them that a planet of mass 5 Me exists between 40 and 75 AU. If Planet X exists its mass should be less and in such an orbit that it does not disrupt the orbit of Pluto.
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In the same month Dr. R.S. Harrington wrote an article called "The location of Planet X", in which he gave several parameters of Planet X and its orbit. Harrington was not the only one who tried to find parameters for Planet X, but he forms a good example. According to him the semimajor axis of the orbit would be 101.2 AU and the mass would be 4 Me. This result is compatible with the result obtained above, so so far it seems that we are getting closer to the exact position of Planet X.
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More articles of this kind were written, but voices against the existence of Planet X rose as well, mainly because all the optical searches (Tombaugh 1961; Luu & Hewitt 1988; Beichmann1987) failed to discover the planet. In 1990, Scott Tremaine published "Dark Matter in the Solar System", in which he discusses the evidence for dark matter near the Sun. He also writes about Planet X in the same article and he states that if there is a Planet X, it must be at least ten times more distant and massive than any other planet. This is so massive that the object would then probably be a brown dwarf, but a brown dwarf would have been detected and that hasn't happened yet.
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The downfall of Planet X
So far we have seen that several astronomers were well on their way to locate Planet X and decrease the area of the sky that would have to be looked at to discover it, when doubt crept in. Then in 1993 Dr. E.M. Standish published an article that washed away nearly all hopes of ever finding the object.
Standish first found out that the observational data used by Harrington and others were inaccurate. Also, by then the masses of the Jovian planets were corrected thanks to better measurements. Using this new knowledge Standish calculated Uranus' orbit again and compared it with its observed orbit and found no significant residuals anymore. He also showed that the residuals in the orbit of Neptune aren't significant and he then concluded that there is no evidence for the existence of a tenth planet. The only reasons astronomers searched for Planet X were the residuals in the orbits of Uranus and Neptune. Standish showed that in fact there are no significant residuals, so no Planet X can be there to be found.
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This of course does not mean that no new heavy objects can be found in our solar system. Since Pluto has no effect on the orbits of Neptune and Uranus, there may be other objects too like Pluto that haven't been found yet. Recently for example, Astronomers from Yale University in the USA discovered a new Plutino (a small Pluto-like object) between Pluto and Neptune. Its radius is 325 kilometers and considered too small to be a planet, so this can not be Planet X, but it shows that the possibility of finding a tenth planet is still open.
Conclusion
After the discovery of Neptune in 1846 the search for a ninth planet to account for the anomalies in the orbits of Neptune and Uranus succesfully ended in the discovery of Pluto. However, this new planet did not seem to have the expected effect on the orbits of the outer planets, so the search went on for yet another planet, Planet X. Despite improvements that were made in the theoretical search, observations to directly find Planet X never yielded any positive results. Astronomers began to doubt its very existence and finally in 1993 Dr. E.M. Standish showed that the possibility of a tenth planet is very small. He used new findings that the Jovian planets have (slightly) different masses than thought before and that the observations of Uranus used before were inaccurate and this way he showed that there are no significant residuals in the orbits of Uranus and Neptune, which were the only reasons to search for a Planet X. Although insignificant, still there are some residuals left, so the possibility of finding Planet X isn't gone completely. Other heavy objects near Pluto may also still be found, which is shown by the recent discovery of a new Plutino between Pluto and Neptune. The existence of Planet X is improbable, but not impossible.
References
Harrington, R.S.(1988).The location of Planet X(Astron. J. 96 4)
Jackson, A.A.,Killen, R.M.(1988).Planet X and the stability of resonances in the Neptune-Pluto system. (Mon. Not. R. astr. Soc. 235 593)
Standish (Jr.), E.M.(1993).Planet X: No dynamical evidence in hte optical observations. Astron. J. 105 5
Tremain, S.(1990).Dark matter in the solar system, from "Baryonic Dark Matter". editors: D. Lynden-Bell & G. Gilmore