--good stuff Jackie!
i went and looked up tarpaulin and turpentine.. and while the dictionary doesn't have as straight line connection... they do seem to share a root!
All in green courtesy of American heratige Dictionary

starting with TARPAULIN
1. Material, such as waterproofed canvas, used to cover and protect things from moisture. 2. A sheet of this material.
ETYMOLOGY: Probably alteration of tar1 + pall1 + –ing2.


which leads us to the 3 root words of tarpaulin--

tar1

PRONUNCIATION: tär
NOUN: 1. A dark, oily, viscous material, consisting mainly of hydrocarbons, produced by the destructive distillation of organic substances such as wood, coal, or peat. 2. Coal tar. 3. A solid residue of tobacco smoke containing byproducts of combustion.

Middle English, from Old English teru. See deru- in Appendix I.
(i didn't copy all the meaning of deru, just one.. if you want more, here is the link)
http://www.bartleby.com/61/roots/IE87.html
deru
6. Variant form *derw-. tar1, from Old English te(o)ru, resin, pitch (obtained from the pine tree), from Germanic *terw-.

Look at that! the tar of tarpauline is from te(o)ru, a word used for resin, pitch... and what is turpentine?
1. A thin volatile essential oil, C10H16, obtained by steam distillation or other means from the wood or exudate of certain pine trees and used as a paint thinner, solvent, and medicinally as a liniment. Also called oil of turpentine, spirit of turpentine. 2. The sticky mixture of resin and volatile oil from which turpentine is distilled. 3. A brownish-yellow resinous liquid obtained from the terebinth.
TRANSITIVE VERB: Inflected forms: tur·pen·tined, tur·pen·tin·ing, tur·pen·tines
1. To apply turpentine to or mix turpentine with. 2. To extract turpentine from (a tree).
ETYMOLOGY: Middle English, resin of the terebinth, from Old French terebentine, from Latin terebinthina (rsna), terebinth (resin), from Greek terebinthin, feminine of terebenthinos, from terebinthos, terebinth tree.


which doesn't lead to deru... but one wonders... two words that sounded similar, and experienced a cross over?

Both tar and turpentine are resins extracted from pine trees. there is no clear connection (sited at this source but is there more? how did the terebinth tree get its name? (is it terebinth tree in greek or is tree redundant? is tere a 'word stem' for 'resin tree'?

turps, is tree resin, as is tar, and tar gives use tarpaulin.. (where did the P of turpentine sneek in?) stolen from tarpaulin? --are turps and tarps closer cousins than this dictionary indicates? tantalizing isn't it?

interesting how words echo throught time.. (like ghosts!)

(i managed not to capture info about pall, its related to a cloth covering (as for the dead), as in pall bearers-- ing as in dancing is pretty well known.)