Type "Nuttall"

This very elegant design is composed of three differentials, coupled by mechanical contraints. First the contraints:

The shafts of wheels 3 and 10 are fixed to the same axle (represented by pseudowheel 4 in the diagram)
The shaft of wheel 7 is mechanically held vertical at all times.

The basic equations (see equation for generic differential):

(a:)	n₂ * rot₂ + n₅ * rot₅ - (n₂ + n₅) * rot₄ = 0
(b:)	n₉ * rot₉ + n₁₁ * rot₁₁ - (n₉ + n₁₁) * rot₄ = 0
(c:)	n₆ * rot₆ + n₈ * rot₈ - (n₆ + n₈) * 0 = 0
(d:)	n₇ * rot₇ + n₈ * rot₈ = 0
(e:)	n₇ * rot₇ - n₆ * rot₆ = 0 (POV to the far right !)

The other constraints are straight forward

(f:)	rot₁ = rot₂
(g:)	rot₅ = rot₆
(h:)	rot₈ = rot₉
(i:)	rot₁₁ = rot₁₂

Composition (POV is from top respectively from far right):

(j:)	(n₇ / n₆) * rot₇ = rot₆ = rot₅ = ((n₂ + n₅) / n₅) * rot₄ - (n₂ / n₅) * rot₂
=>	rot₄ = (n₅ / (n₂ + n₅)) * (n₇ / n₆) * rot₇ + (n₂ / (n₂ + n₅)) * rot₁
(k:)	(n₇ / n₈) * rot₇ = (-1) * rot₈ = (-1) * rot₉ = (n₁₁ / n₉) * rot₁₁ - ((n₉ + n₁₁) / n₉) * rot₄
=>	(-1) * rot₄ = (n₉ / (n₉ + n₁₁)) * (n₇ / n₈) * rot₇ - (n₁₁ / (n₉ + n₁₁)) * rot₁₂
(l:)	[(n₅ / (n₂ + n₅)) * (n₇ / n₆) + (n₉ / (n₉ + n₁₁)) * (n₇ / n₈)] * rot₇ + (n₂ / (n₂ + n₅)) * rot₁ - (n₁₁ / (n₉ + n₁₁)) * rot₁₂ = 0

To make it "south-pointing" requires, that if rot₁₂ is held 0, rot₇ has to be exactly one turn when wheel 1 of diameter d₁ covered one full circle (centre at wheel 12's contact to the ground, radius equal to track width t):

(m:)	d₁ * rot₁ * PI = 2 * t * PI
(n:)	rot₇ = 1
(o:)	rot₁₂ = 0
(l=>)	rot₁ = (-1) * [(n₂ + n₅) / n₂] * [(n₅ / (n₂ + n₅)) * (n₇ / n₆) + (n₉ / (n₉ + n₁₁)) * (n₇ / n₈)]
(m=>)	d₁ = 2 * t / \| rot₁ \|

The same method can be applied to calculate the required size for wheel 12: If rot₁ is held 0, rot₇ has to be exactly one turn when wheel 12 of diameter d₁₂ covered one full circle (this time centre at wheel 1's contact to the ground, radius equal to track width t):

(p:)	d₁₂ * rot₁₂ * PI = 2 * t * PI
(q:)	rot₁ = 0
(l=>)	rot₁₂ = [(n₉ + n₁₁) / n₁₁] * [(n₅ / (n₂ + n₅)) * (n₇ / n₆) + (n₉ / (n₉ + n₁₁)) * (n₇ / n₈)]
(p=>)	d₁₂ = 2 * t / rot₁₂

Given Parameters (Your choice !)

Reference	Size	Description
2	n₂ = [teeth]	Bevel gear, fixed to wheel 1
3	n₃ = [teeth]	Bevel gear, freerunning on vertical shaft
4	-	Main axle, only fixed to the vertical shafts of wheel 3 and 10
5	n₅ = [teeth]	Bevel gear, freerunning on main axle
6	n₆ = [teeth]	Bevel gear, fixed to wheel 5
7	n₇ = [teeth]	Bevel gear, fixed on vertical pointer shaft
8	n₈ = [teeth]	Bevel gear, freerunning on main axle
9	n₉ = [teeth]	Bevel gear, fixed to wheel 8
10	n₁₀ = [teeth]	Bevel gear, freerunning on vertical shaft
11	n₁₁ = [teeth]	Bevel gear, fixed to wheel 12
T	t = [length units]	Track width

Derived Parameters

Reference	Size	Description
1	d₁ = [units of t]	Roadwheel, freerunning on axle
12	d₁₂ = [units of t]	Roadwheel, freerunning on axle