Surface prep is supremely important.
The ivory should be sanded and polished to a scratchless surface. I really like the 3M sheets that come in a color coded packet. They go from 400 to 8000 grit. They do not stain the ivory with SC grit as the wet-or-dry papers do. I have found nothing that does ivory better.
Before starting the scrim, look at the surface with a magnifier - It should look slick as glass before the scrim work starts.
Sealing with CA or lacquer before the last sanding /buffing sometimes helps if the ivory is porous. After the scrim is done, the pigment is applied. It is important that the scrim work does not get oil or anything in the pricks and scratches , as this will keep the ink out. Brush the scrim off and blow it out before applying the ink. Drawing inks, paint, or any other permanent pigment will work. India ink is my preferred pigment. More on that later. Apply the pigment and make sure it is worked in to the scrim. Wipe off the bulk of the excess gently. Let dry, and rub/buff with something that won't remove any of the ivory, but will rub off the ink/paint. I use a barely damp cotton cloth first, and then a piece of brown paper bag as the rubbing material. I then gently buff with pink and white 3M sheets ( 4000 and 8000). I find 4/0 steel wool too abrasive, as well as it tends to pull some of the ink out of the scrim work.
A sealer coat of artist lacquer, like is applied over charcoal drawings, will help preserve the scrim, but with India ink, it is not really needed.
The thing to always remember is that any scratch or crack in the ivory will take in the ink/paint. Either mask off or seal these areas before starting....or expect them to show in the final art work. Sometimes this can be used to advantage by taking a natural feature and incorporating it into the design. For instance, I had a piece of walrus with tiny age cracks along it. They were small and all running in the same direction, but there were a lot of them. I made a scene of a pod of walrus in the ocean and some ice floes. The lines became waves and ripples, and the edges of the ice......and looked perfectly natural. I was careful to apply the ink to the scrimmed area only, so as not to fill the cracks outside the artwork perimeter.
BTW, China white is a good method of coating the ivory and sketching on it before the scrim.
Now, it wouldn't be a professor Stacy post without the science talk:
Scrimshaw works so well on ivory because the surface is hard and dense. The ivory can be sanded to a very smooth surface and then buffed to be even smoother, and harder. By nature ivory is waterproof, so it does not allow the wet ink to "penetrate" the surface, and get a grip. The second part of the scrimshaw equation is the physical nature of fracturing in ivory. It fractures with a splintery and rough fracture. When you use a stylus to scratch or prick the smooth hard shell that was created in the buffing, it makes a porous groove or pit with tiny spikes sticking out. These are fractures. Look at a scrim mark with a powerful magnifier or microscope and you will see that it is indeed very rough and grippy. When the scrim is done, there are thousands to millions of these pits and lines...all ready for some ink to try and invade the ivory. The ink is applied and allowed to pool in these areas. It is important to not completely wipe off the excess before it is dry. The pigment attaches itself in these microscopic jagged canyons and valleys. When dry, the surface is rubbed with a fairly non abrasive material. This creates heat by friction, dislodges the dried material on the slick surface where it has no grip, and drives it into the jagged grooves and dots even harder. A little gentle buffing with a flannel cloth or extremely fine polish cloth will give the ivory a smooth oily luster and the ink a bold contrast. Done with a long lasting permanent ink, like India ink, this will last for centuries.
While on this science field trip, lets look at that India ink and see why it works so well...and has for several thousand years.
India/Chinese ink is a simple ink. It is pure water, carbon, and a binder. What makes it attach to paper (and ivory) so well is the source of the pigment and the type of binder. The pigment is mostly pure carbon. Lamp black, pine charcoal, and pine wood ash are mixed to make a sticky black pigment, a little pitch/tar/shellac is often added, and this is mixed with water. These pigments and binders don't really dissolve in the water, but float around in it as minute balls of sticky black goo. It makes an organic mix that dries quickly, and bonds tightly. It also has an affinity for organic materials, so it bonds well to cloth, paper, wood, ivory, (and especially well to carpets, shirts, pants, and skin 
) but will wipe off china and glass. It invades and coats every surface of the scrim marks jagged fractures, getting a very strong grip. Once the water has evaporated, the oils, carbon, and resins will cure into a hard waterproof material similar to asphaltum. It can be cleaned up fairly easily, and when fully dry, has great wear resistance and durability. Rubbing hard will melt this asphaltum material and allow removal from the high spots, and pack it into the grooves. A brown paper bag works well as a rubbing material because it is softer than the ivory, porous so it carries away the excess pigment, absorbent so it absorbs any residual moisture, and the rough surface makes friction when rubbed hard. When the ivory and ink are dry and cool, they take a nice polish by a more gentle buffing with a soft cloth.
