The following recipe was first shared by Roland Cadle, but then was found in an 1876 chemical recipe manual by Charlie Kaufmann, who is a chemical engineer.
Mix one part lead oxide powder, with two parts powdered quick lime. This is done dry, stir the powders together so they are fairly uniformly blended. Next, mix 1-1/2 to 2 teaspoons of commercial lye powder (such as Red Devil brand, or any other brand sold for cleaning drains) with one cup of warm water. The warm water helps the lye to go into full saturation. Do this part in a well ventilated area, as fumes will be formed when the lye is added to the water. Do not breathe the fumes directly, as it is a "breathtaking experience".
Once the lye is in solution, then take a very small amount of the lye water and mix it into the powdered lead oxide/quick lime mixture to form a paste. This paste can then be applied to the surface of the horn in the area that you wish to color. Allow the paste to dry and crust over on the horn. (10 - 15 minutes) At this point, depending on the strength of the lye solution, your color reaction should be well along. You may find that you need to "re-charge" the reaction slightly by adding a little of the lye solution to the already applied paste. This sometimes seems to drive the reaction along and make the color darker and more vivid.
At this point you can wash the crusted paste off the surface of the horn to see what you have colorwize. If you like it, you are done, if not, re-do the process to get the level of color you want. The intensity of the color on the horn does tend to darken over the next few hours, and will especially darken if you heat the horn during the dyeing of the horn body.
This is not an exact science. You can vary the amount of lye in the water solution, and you can vary the ratio of the two powders.