What This Coin Looks Like (Obverse, Reverse, Mint Mark Location, Special Features, etc.): There were two major varieties of the 1839 Liberty Head $10 Gold Eagle. This variety used the older style liberty head on the obverse - known as the Type of 1838 - and also larger letters on the reverse of the coin (as depicted above).
The 1839/8 (Type of 1838) is easily distinguishable from the 1839 (Type of 1840) design. The old style liberty head is tilted downwards making the date appear more to the left of the liberty. The old style liberty features a narrower and sharper curve at the bottom-right portion of the liberty bust and also the ear is covered a bit more and in a different style. The new style liberty (Type of 1840) is aligned with the date and the bust is tilted upward more. The bottom-right curve on the liberty bust is shorter and wider - with less hair covering the ear and in a different style. This is the design used on all liberty head $10 eagle coins in 1840 and subsequent years. The difference between the Type of 1838 vs Type of 1840 is depicted below (Source of example images is courtesy of Heritage Auction Galleries):
In addition: The older "Type of 1838" features slightly larger and taller letters on the reverse of the coin - while the newer "Type of 1840" coin features smaller letters. The difference between the Large Letters vs Small Letters is depicted below in the example comparison image (Source of example image is courtesy of Heritage Auction Galleries):
Lastly: The 1839 "Type of 1838" features a "9 Over 8" overdate resulting from an 1839 date struck over the older 1838 date. Subtle parts of the outer loops on the right side of the underlying "8" can be seen just to the right of the "9". Depicted below is an example closeup image of the 1839/8 overdate:
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