Tandem cycle sprockets

Tandem bicycle chain centre sprocket sizes differ why?
Note We all know they have a big one at the pedals and a little one on the wheel,
that’s not the point, on a tandem you have two chain wheels side by side but different sizes. why?

I’m not sure I understand your question. Lots of bikes (not just tandems) have two or more chain rings of different sizes side by side, they are for changing the gear ratio. A tandem bike will have yet another chain ring that is on the opposite side to the conventional drive side and is connected to the chain ring of the front rider (or “pilot”). The chain rings connecting the pilot to the stoker just need to be the same size as each other so that the two rider’s pedalling stays in sync.

In the example below, there are three chain rings connected to the final drive chain to give the riders a wide range of gearing options. Then there is another “chain ring” on the opposite side (darker grey in colour in this case) that is purely to connect the pilot to the stoker and has no effect on gearing.

http://www.eriksbikeshop.com/Assets/product_images/PR3C14280_mlg.jpg

And below is a non-tandem bike with two chain rings.

http://dfp2hfrf3mn0u.cloudfront.net/234/234081_65525_tif_zoom_1.jpg

You understood the question… the reason for the question being asked is that when views from directly side on and from a distance, it looks like the two riders would be pedalling at different speeds and mechanical advantage.
No, the chain connecting the two riders cranks maintain sync , speed and the same mechanical advantage/gear ratio (foot speed to wheel speed)

The chain ring at the second drivers crank that isn’t the match for the first riders chain ring is one to drive the rear wheel… this is just sized to give the correct gear ratio given the sizes of everything else… (pedals crank length, rings at rear wheel, size of rear wheel.)

The sync chain sprockets are on the left side of most tandems. (exceptions are not uncommon *)

This chain only links the two crank-sets and keeps them synchronized (hence the name) This chain transfers power from the captain’s (front) bottom bracket to the stoker’s (rear) bottom bracket. the right side of the stoker’s crank-set is nearly identical to a solo bike. (greater range is desireable on a tandem).

The size of the sync chain chain rings is normally 40 (+/- 2) teeth on road tandems. This is chosen as a compromise between efficiency and chain durability, which favors large chain rings, and weight and ground clearance which favor smaller rings. MTB tandems will use smaller sync rings to gain ground clearance.