What is the irregularity in Uranus' orbit that is caused by Neptune? I carefully read the Wikipedia article Discovery of Neptune, and I don't get what the irregularity of Uranus orbit was that lead to the discovery of Neptune. Years ago, I watched some educational film that schematically showed the irregularity as follows. When the two planets were "near" each other Neptune's gravity would temporarily pull Uranus out of its orbit and then Uranus would be somehow pulled back and continue orbiting along an elliptical orbit as if nothing happened. So Uranus' orbit would be perfectly elliptical except some rather short arc where it would be bent outwards.
This doesn't make sense - if there's some external force that "pulls" (even slightly) Uranus out of its orbit how would it get back into orbit? I suppose its orbit must have permanently (yet slightly) changed. So as time passes, successive deviations must accumulate each time Neptune passes Uranus and Uranus' orbit must get bigger (further from the Sun) and since Neptune is also attracted to Uranus by the very same gravity Neptune orbit must get smaller (closer to Sun).
What exactly is the irregularity and why is Uranus' orbit not permanently changed each time Neptune passes by?
 A: Orbits are funny things really The key thing is the distance from the object that they are orbiting, and the speed at which they are going. 
Spacecraft routinely speed up/ slow down their orbital speed, to affect where they are orbiting around a particular spacecraft.
So, the real answer is, Neptune pulls Uranus closer when it is ahead of it in orbit, and slows it down when it is behind it in orbit. The net effect is to almost cancel each other out. But you are right, over a very long period of time, it can have an affect. The same is true of all planets, they affect each other over a long period of time. It actually was part of the irregularity discussed previously (See the article The Discovery of the Outer Planets).
A: Understanding planetary perturbations is easier when you imagine them in the correct frame.
Imagine a frame centered on the real planet. One axis points away from the sun, the other is perpendicular to the first, in the "leading" direction of motion. Distances are measured in km or in AU. Each axis corresponds to one component of the perturbation: one is a perturbation in the radius vector, this other in heliocentric longitude.
In this frame, one can plot the trajectory of the unperturbed planet relative to the real one. 
On such plots, one sees that:
- on short timescales (tens of years), the unperturbed planet loops more or less on a non-closed ellipse that is centered either "before" or "after" the real planet. In these "ellipses", perturbations in radius vector are about half of those in longitude (in km)
- on longer timescales (centuries), the center of these loops slides in a cyclical manner between extremes in the "before" and "after" positions.
For the solar system, these plots reveals three planetary couples where partners counter each other:
- Jupiter and Saturn,
- Uranus and Neptune,
- the Earth-Moon barycenter and Mars.
When one member of each couple is ahead of its unperturbed position, the other member is late, and conversely. Historically, this was first noticed with Jupiter and Saturn. The irregularities were noticed during the 17th century, and their inversion, during the 18th century. No one was able to explain what was happening until the problem was solved by Laplace in a memoir to the Académie des Sciences in Paris.
Perturbations between Jupiter and Saturn are cyclical and have a periodicity of 883 years. This is called the "great inequality".
The explanation goes along these lines given on the following link, paragraphs 196-198, with a bonus about Uranus and Neptune paragraph 199.
http://books.google.fr/books?id=YnkMAAAAYAAJ&dq=great%20inequality%20883&hl=fr&pg=PA209#v=onepage&q=great%20inequality%20883&f=false
A useful source with plots: Jean Meeus, Mathematical Astronomy Morsels III, chapter 30
A: Since gravitational acceleration is a vector, we can sum the attractions from the sun and Neptune on Uranus into a single acceleration. We can then break that single acceleration down into two components: one towards the sun, As, and the other in the direction of motion, At.
We will now consider the orbits of Uranus and Neptune around the sun, and we will consider motion relative to Uranus. As Uranus catches up to Neptune, there will be a time where Neptune is straight ahead, and the angle sun-Uranus-Neptune is 90 degrees. Call that point QE (eastern quadrature). Later, Neptune will pass directly opposite the sun, as seen from Uranus. Call that point O (opposition). Later, Neptune will be directly behind Uranus, with the angle sun-Uranus-Neptune again 90 degrees. Call that point QW (western quadrature). After a long time, Neptune will keep being left behind until it is directly behind the sun as seen from Uranus. Call that point C (conjunction).
Now consider At. As Neptune goes from C to QE to O, At is always in the direction of motion of Uranus, and reaches a maximum between QE and O. As Neptune goes from O to QW to C, At is always against the direction of motion of Uranus, but with the same magnitude as it had going from C to QE to O. Hence Uranus will be accelerated for half the time and deccelerated half the time due to At, and the net acceleration will be close to zero.
Now, consider As. While Neptune goes from QE to O to QW, As will be decreased, causing Uranus to move further away from the sun. While Neptune goes from QW to C to QE, As will be increased, pulling Uranus towards the sun. Of course, the magnitude of this increase will be a lot less then the earlier decrease, because Neptune is much further away, but it acts for a much longer time! So again, the net result is to put Uranus back where it had been originally. 
In summary, the effects of Neptune on Uranus can be easily detected for parts of the orbit, but the net result after a whole orbit is very close to zero. As Pearsonartphoto pointed out, the net result is not exactly zero because of second order and higher terms. But so far, celestial mechanicians have found that, while these higher order terms can cause periodic effects lasting tens to thousands to hundreds of thousands of years, all such effects seem periodic over the long term. After billions of years, chaotic forces dominate anyway.
