Differential Ring Gears

Gears most commonly have teeth on the outside edge of a disk. The teeth of a gear can also be placed on the inside edge of a ring.
 
The relationship between the small gear and the large gear stays the same whether the teeth are on the inside or the outside of the large gear. Moving the teeth to the inside doesn't change the relative movement between the two large gears either.
 
The large orange gear advances by one tooth each time the small orange gear completes one orbit. After 25 orbits of the small orange gear, the large orange gear rotates once. After one orbit of the small orange gear, the large orange gear rotates 1/25, or 0.04 times.
 
The following formula calculates this reduction factor:
 
1 - ( (large blue ÷ small blue) × (small orange ÷ large orange) )
1 - ( (24 ÷ 12) × (13 ÷ 25) )
1 - ( 2.0 × 0.52 )
1 - 1.04
-0.04
 
A negative reduction factor indicates that the ring gear rotates counter to the orbit of the small gear.

Differential Movement

In the earlier example, both the large orange gear and the large blue gear were rotating. The large orange gear was rotating slightly faster than the large blue gear.
 
In this example, the large blue gear is no longer rotating. When the large blue gear is stopped, the small blue gear can no longer rotate in place. Instead, the small blue gear rotates and orbits around the large blue gear.
 
The small blue gear is turning at the same rate as before. After each two rotations (and one orbit) of the small blue gear, the dark wedge on the small blue gear aligns with dark wedge on the large blue gear. (The wedge is visible on the lower right-hand side of the large blue gear.)
 
As the small blue gear rotates and orbits, the small orange gear rotates and orbits with it. The small orange gear causes the large orange gear to rotate at a rate which is the difference between the earlier rotational speed of the large blue gear and the rotational speed of the large orange gear.

Relative Movement

With the gear pairs aligned one above the other, the difference in rotational speeds becomes more visible. Tracking the dark wedges on the surface of the gears enables you to see the relative movement of the rotations.
 
The small blue and the small orange gears turn at the same rate, so the dark wedges on the two gears stay aligned. The large gears turn at different rates, so the dark wedge on the large blue gear and the dark wedge on the large orange gear move relative to one another.
 
The large orange gear is rotating slightly faster than the large blue gear. Each time the large blue gear rotates once, the large orange gear rotates a bit more than once. The dark wedge on the orange gear slowly moves ahead of the dark wedge on the blue gear.

Basic Gear Movement

When the teeth of gears are meshed, the movement of a tooth on one gear results in the movement of a tooth on the other gear. The teeth of each gear move together at the same rate, but when the gears have different numbers of teeth, the gears rotate at different speeds.
 
In the pair of blue gears, the large gear has 24 teeth and the small gear has 12 teeth. Each time the small gear makes one complete rotation, its 12 teeth engage with 12 teeth on the large gear.
 
The large gear has 24 teeth, so after turning by 12 teeth, it hasn't made one complete rotation, it's turned only 12/24, or 1/2 of a rotation. It takes two turns of the small blue gear to turn the large blue gear one time.
 
In the pair of orange gears the large gear has 25 teeth and the small gear has 13 teeth. Each time the small gear turns two times, the large gear turns one time plus one tooth. Because of the additional movement of this one tooth, there is a small difference in the rotational speeds of the two large gears.