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Looking at the spectrum of where jet aircraft put their engines, there are examples of almost every place that could conceivably used to carry a jet engine:

Notably absent, however, are jets with engines having anything to do with the horizontal stabilizer.

Why aren't there any jets with engines mounted on/in their horizontal stabilizer?

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    $\begingroup$ Related: Would there be any benefit in placing the aircraft engines on the elevator (end-tips)? $\endgroup$ – fooot Oct 17 '18 at 21:37
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    $\begingroup$ Because we want the stabilizer to have smooth air, hence away from anything, including the engine. $\endgroup$ – user3528438 Oct 18 '18 at 3:04
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    $\begingroup$ I think some fighter jets like the F-14 would come pretty close. But in general, it's just a matter of structural strength: you'd have to make the stabilizer much stronger to handle the thrust of the engine. $\endgroup$ – jamesqf Oct 18 '18 at 3:29
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    $\begingroup$ Your linking game is en pointe. Especially the first example! $\endgroup$ – Jamiec Oct 18 '18 at 6:05
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    $\begingroup$ It may not be what you are looking for, but engines are placed on the stabilizing part of concorde's delta wing. $\endgroup$ – qq jkztd Oct 18 '18 at 9:37
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There are two configurations I can think of.

1. The jet mounted on the horizontal stabilizer (above, below, or the tip)

Unlike the vertical stabilizer, the horizontal stabilizer is trimmable (THS). It's constantly moving, which means when it moves, so would the thrust line. A kind of thrust vectoring that you don't want. For example, trimming for slow flight during approach, the thrust line will point down and aft of the CG, which will in turn point the nose further up, this is a kind of a destabilizer. Even if the angle is not too big, the changing thrust line will always act against the intended trim. And lastly with the engine constantly on the move, the blade tip clearances will need be bigger, which will in turn reduce the engine's efficiency.

The required strength issues that result in a heavier weight are also a factor. This also impacts the loading flexibility; a tail heavy plane will reach a point where you can't put as many passengers in the back without adding a forward ballast.

2. The jet mounted at the root and doesn't move with the THS

While certainly achievable, like the L-1011's number two engine wrt the vertical stabilizer, the issue here is not why aren't the jet engines there, it's why the THS is not there.

The horizontal stabilizer of a plane like the DC-9 (and its extended family) is purposefully placed above the tail, creating a T-tail. With the tail swept back, the top of it is farther from the plane's center of gravity, giving the THS a bigger moment arm, and in turn allowing a smaller THS, and the other benefit that comes out of that is the end-plate effect of the T-tail, which also permits a shorter fin for the same control authority.

You'll also notice buried engines are of the older small-diameter type, except for large planes like the L-1011, but still that makes maintenance access a lot harder, than an exposed engine.

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