The wire braces of a yard are running to the separate drums, which are fixed on one shaft. Both ends are rove on the smallest diameter of the drum, one is attached on the front side, the other on the backside of the drum. If the yard is braced up, one drum is taking in the wire while the other drum is paying out.

        If the yard is braced, let’s say on starboard's tack, the angle between the brace and the yard-arm on portside increases, on starboard side this angle decreases, so the brace is running more alongside the yard. Or as Jarvis puts it: the sum of the both braces of a sharp braced yard is smaller then the sum of the braces in square position.
This is the reason that on portside the length of wire to be taken in is increasing, while on starboard the length of wire paid out is decreasing with an increasing brace angle. Therefore coning drums are needed. The exact shape of the conical drum is found empirical by bracing the yard about several times.

        One winch can serve three yards and, as said before, this can be every other yard to brace the complete mast, or just the lower three yards. In the calculations that follow, only the second option is worked out, because bracing the complete mast was not done in the declining years of the sailing trade.
For optimal sailing performance, when close hauled to the wind, the yards have a little twist or fanning. This fanning is required due to the fact that the true wind at sea level is less strong then at royal-mast height. This means that the apparent wind at sea level is coming in more forward then the apparent wind at height of the royal. Therefore the lower yard has to be braced up the sharpest.
This fanning also has to be established on the winch. When sailing close to the wind, the lower topsail yard is braced up 3 to 5° less then the lower yard. The upper topsail yard again 3 to 5° less then the lower topsail yard, and so on.

I have just learned that the 5-mast bark 'France' was equiped
with ten-drum winches to accomodate all the yards of a single mast. This drawing is
taken from a study about those winches by Jan Huerkamp, Navcon rigging company.

        In general the brace is rove as follows: the standing part of the brace is in the side of the ship, on the rail or on a bumpkin. From there the brace runs up to the brace block on the yardarm and from there to a guiding block under the top or crosstrees. From there it runs to the winch. This end is the running part.
The standing part of the brace is put on a tackle. Stretch in the wire and little differences between the yards can be adjusted by this tackle.

A haliard winch manned at the fourmasted bark Krusenstern.
Provided by David McGovern.

Note: Another application for the use of conical drums is on halyard winches and leeboard winches, but for different reason.
Halyard winches and leeboard winches can be equipped with conical drums, for the simple fact that when a yard or a leeboard nears its top position the strength needed to wind them up higher, increases. For a leeboard that is clear enough, the higher it gets, the less the upward force of the water is co-operating. For the yard it is a bit different, but I guess it has to do with the sail which streches more and more and will get more and more pressure of the wind. So they will become heavier to lift. If the diameter of the winch drum decreases, the couple of forces will also decrease and it will take less strength to put them up completely.

I don't know who made this drawing, but you will find it everywhere. But now I do know why they drew it with the braces on the large side of the drum. It is for the same reason as I was mentioning above, the reason of the reducing arm of forces. The person, who made this drawing, was taking the principal of the coning drums for granted. 
I found prove of that in the Dutch Maritime Encyclopaedia of 1971. At the item "Jarvislier" it states:
(…) The winded up lee braces are always under greater tension then the simultaneously unwinding windward braces, so that on the one side there is hardly any slack, while on the other side the bracing up is not hindered by the 'luff' braces. When bracing up to close-hauled, the drum diameter will be at its smallest and therefore the reduction and the force employed will be the greatest. (…)  No further comment.

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