One of the best know publications of Douglas Hill is Extended Techniques for the Horn: A Practical Handbook for Students, Performers and Composers. This is an extremely comprehensive resource.
On lowering an open note to a stopped note
Perhaps in reflection of that comprehensive study his section on the topic in Collected Thoughts on Teaching and Learning, Creativity, and Horn Performance looks not only at hand stopping but also the common extended technique of lowering an open note to a stopped note.
The next level of concern comes when a composer wishes for the sound to slide in and out of stopped and open horn through various glissandi effects…. The performer and the composer must realize that to move the right hand gradually over the bell does not raise the pitch a half step. It lowers the pitch. The interval of change is totally dependent upon the distance to the next lower harmonic of that particular fingering.
Hill shows that in the lower range for example a written middle C played on the F horn would be lowered to an Ab below that note by stopping the note with the hand, in other words, lowering it to the half step above the next lower harmonic. This type of hand stopping is closely related to the hand stopping used on the natural horn.
Five common problems
Hill includes suggestions on five common problems related to production of stopped notes with suggestions to correct them
1. Producing a fully closed and compressed sound…. Be sure your thumb is pulled back alongside the index finger…. The cover should be thought of as a door hinged at the middle, simply covering the oval-shaped opening, with little or no force. (Do not try to stuff the bell with your fist, thinking of it as a circular shape.)
2. Projecting an appropriate volume. The inside of the mouth should be widened into a larger vowel formation similar to playing lower. The air projection should be more focused and forceful as you blow against the added resistance. It should feel like you are blowing a full ff if you wish to project a mf volume (f=mp, etc.).
3. Finding the pitch. Aim with your ear and the vibrational sensation at the lip for the actual sounding pitch, not the one you are fingering….
4. Accuracy in the upper register on the Bb side of the instrument. The use of the stopped horn hand position is consistent for the F horn. However, there are flat fingerings which can be used on the Bb side that can give you greater security….
5. Control or simply production of a stopped sound below written middle C. Here we run into a varied set of problems. The most common is an inadequate cover with the right hand….
Continuing with point five, the other elements include the forceful air stream and hand size in relation to bell size. In regard to that Hill mentions one solution being a modified rubber kitchen glove on the right hand. “Cut off the finger tips to the second knuckle, the thumb tip down to the first knuckle, and the wrist/palm area up to near the base of the thumb.” He reports that “Wearing such a device does not overly distort the neighboring open notes and does broaden a small hand and fill the holes between the fingers.”
Echo horn, etc.
Hill also discusses the related topics of echo horn, half stopping, and three-quarter stopping.
These three names refer to basically only one technique. The desired and expected sound is usually one of a distant echo or a veiled whisper.
He notes part of the complexity of this effect is that your hand position will vary and needs to be more closed for higher notes than it would need to be in the lower range. On the positive side however, “It actually works better on the Bb side in the top octave since the harmonics are farther apart, thus providing a more characteristic echo, rather than a stopped horn sound.”
Returning to his earlier publication, Extended Techniques for the Horn: A Practical Handbook for Students, Performers and Composers, this book and CD have much to offer on extended techniques and is very highly recommended for additional study for those with a special interest in the topic.
A pet peeve, and a bit more on using flat fingerings to play stopped in tune
To close, there really is more to be read on the topic of stopped horn out there, ranging from the very practical to writings that are so theoretical that they are very hard for a mere mortal to decipher. A related pet peeve of mine: I have yet to find a theoretical article that could explain why on a large bell horn I can easily play in tune stopped, but on a smaller bell instrument I tend to be sharp and have to resort to the use of flatter fingerings (the Horn Harmonics device is useful for finding these). Students with small hands tend to be sharp no matter what and finding flat fingerings is a lifesaver (explained further at the link above, but in short for example if your stopped B on the third space is sharp, try fingering it 0 on the F horn, a fingering for Bb that is about a quarter step flat, a harmonic that can be used for quite a number of notes in the middle of the horn). When you get down to it as a performer, practical reality is more important than beautiful theories that break down with the reality of variations of bell and hand sizes.
You need to own a stop mute
The final thing to add being that every advanced horn player needs a good stop mute, it is a life saver! Tone production is much more consistent and powerful, with much better intonation.
There are some good standard choices out there (Trum-Cor, Alexander, etc.), and for those interested in the latest trends check out this review of the Woodstop and Ion Balu stop mutes, an article that also includes some practical notes on stopping on the high F side on a descant or triple horn.
When we return the Hornmasters series will continue with a group of articles related to performance anxiety.
Back in 2007 Horn Notes Edition released Playing High Horn by John Ericson. 
Understanding Range Development
Rachel asks:
To get a pulse on what our readers might be thinking in terms of current affairs in the American symphony orchestra scene, a poll was posted last Wednesday with one simple question:
Farkas expands on this general topic in The Art of Musicianship. Another publication by a horn player that covers some of the same general information in a practical and concise format is The Rules of the Game by Christopher Leuba, which I also recommend the reader looking for more information to consult. 
The list of troubled U.S. orchestras seems to get longer and longer. It feels as if a dam has burst and we are being hit by wave after wave of negative publicity.