How archaic and fun!
I whipped out some Google-fu and found the following for you:
Tincture of Capsicum
You can actually buy this on Amazon: Cayenne Capsicum Tincture 2 Ounces. It's available other places, but I saw prices as high as 2x this. (9 ml ~ 0.3 oz)
Essence of Ginger
This is from a late 19th century Jamaican cookbook (Classic Jamaican Cooking: Traditional Recipes and Herbal Remedies).
- Three ounces of freshly grated ginger
- Two ounces of thinly cut lemon-peel
- Two pints of brandy or proof spirit (white rum)
Just combine and let sit for 10 days, shaking well.
Essence of Lemon
This can also be purchased. You can likely find some lemon extract in your grocery store, if not here is some on Amazon: Flavorganics Organic Lemon Extract, 2-Ounce Glass Bottles (Pack of 3) - note that it's 3 bottles.
Solution of Burnt Sugar
This isn't just typical caramelized sugar, it's burnt sugar. Sugar that has been heated to the dark caramel stage of 370 F to 400 F (188 C to 204 C). The sugar at this point has lost most of it's sweetness, smells burnt, and tastes kind of bitter. It's commonly called caramel color these days. This is what is gives Cola's their dark color. In the small amount of 25ml and in the presence of 2.5 lbs of sugar (wow!) you won't taste it.
My amazon-fu fails here, but have found it available on this site: http://www.spiceplace.com/mccormick_caramel_color.php
It might be simpler to buy this than attempt making your own. Caramel coloring can be a little hazardous to make, it's easy to go too far and burn your sugar, and dumping any quantity of water into 400 F sugar is a sketchy experience. Here is a how-to if you desire though:
http://www.ehow.com/how_5673239_make-caramel-food-coloring.html
Tartaric Acid
This too can be purchased - Tartaric Acid 2 oz. (56 gr.)
Please note that tartaric acid is not the same as cream of tartar. Cream of tartar is an acid-salt in which the tartaric acid is partially neutralized. This site suggests you can use a 2:1 substitution ratio of cream of tartar to tartaric acid. I'm not sure if this will make a difference in your drink because I've never used cream of tartar in such a way or quantity before.
There is also the possibility that a recipe of this age simply used the term tartaric acid to refer to cream of tartar. Who knows?
Here is a good list of choices for substitutions: http://foodallergies.about.com/od/cooking/p/cookingwoonions.htm.
The thing I found interesting though, is what it says about Garlic and Onion powder:
Garlic powder and garlic salt are made from dehydrated, finely ground garlic. Onion powder and salt, likewise, are made from onions. These products are not suitable for anyone with a true allergy to either of these foods. However, some people who do not have true allergies to these foods and avoid garlic and onions due to gastrointestinal upset find that these products do not aggravate their symptoms, especially if used in small amounts.
So the question becomes, is it an actual allergy? You may want to check with the doctor on that one.
Here's their list of substitutions:
Fennel has a licorice-like taste but onion-like texture. Try it with chicken or fish.
Celery is among the most common aromatics.
Bell peppers are often used in Cajun cooking. Green peppers and celery are a good base for rice dishes or savory stews.
Carrots are used as an aromatic in French cooking in combination with celery.
Celeriac, or celery root, is the knobby root of one variety of celery. Peeled and diced, it can be used as an aromatic in sauces or
stews.
Peppercorns: white, pink, or Szechuan pepper can add different flavors to your cooking.
Cumin's distinctive taste that may work well in some recipes, especially where garlic is used raw.
Horseradish, freshly grated, can add some of the pungent notes you might otherwise lack.
Ginger and galangal have distinctive flavors but may be useful in stir-fries as aromatics.
Best Answer
You can very likely buy frozen lemon juice or juice concentrate, even if you can't buy fresh lemons. (This is different from frozen lemonade concentrate.)
This is what we do often when we're making large batch lemonade; while it's not quite as good as fresh lemon juice, it's usually close enough that it's not noticeable.
For example:
It's kept in the freezer section. That jar is roughly seven lemons' worth of juice; each one makes something like 2-3 liters of lemonade, depending on your strength requirements. I usually put a few sliced fresh lemons in with the pitcher after that.