Type "Lanchester"

Central element of this design is a differential put on end. Compared to the generic design, wheel 6 is "left hand side (2)", wheel 9 is "right hand side (1)" and wheel 8 is the "difference (4)". The adapted equation reads

(a:)	n₉ * rot₉ + n₆ * rot₆ - (n₉ + n₆) * rot₈ = 0

The other constraints are straight forward

(b:)	rot₁ = rot₂
(c:)	rot₃ = rot₄
(d:)	rot₁₁ = rot₁₂
(e:)	rot₁₃ = rot₁₄
(f:)	n₂ * rot₂ + n₃ * rot₃ = 0
(g:)	n₄ * rot₄ + n₅ * rot₅ = 0
(h:)	n₁₀ * rot₁₀ + n₁₁ * rot₁₁ = 0
(i:)	n₁₂ * rot₁₂ + n₁₃ * rot₁₃ = 0

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

(j:)	rot₆ = rot₅ = (-1) * (n₄ / n₅) * rot₄
	= (-1) * (n₄ / n₅) * rot₃
	= (-1) * (n₄ / n₅) * (n₂ / n₃) * rot₂
	= (-1) * (n₄ / n₅) * (n₂ / n₃) * rot₁
(k:)	rot₉ = rot₁₀ = (-1) * (n₁₁ / n₁₀) * rot₁₁
	= (-1) * (n₁₁ / n₁₀) * rot₁₂
	= (n₁₁ / n₁₀) * (n₁₃ / n₁₂) * rot₁₃
	= (n₁₁ / n₁₀) * (n₁₃ / n₁₂) * rot₁₄

To make it "south-pointing" requires, that if rot₁₄ is held 0, rot₈ has to be exactly one turn when wheel 1 of diameter d₁ covered one full circle (centre at wheel 14's contact to the ground, radius equal to track width t):

(l:)	d₁ * rot₁ * PI = 2 * t * PI
(m:)	rot₈ = 1
(n:)	rot₁₄ = 0
(k=>)	rot₉ = 0
(a=>)	n₉ * 0 + n₆ * rot₆ - (n₉ + n₆) * 1 = 0
(=>o:)	rot₆ = (n₆ + n₉) / n₆
(j=>)	rot₁ = (-1) * (n₅ / n₄) * (n₃ / n₂) * (n₆ + n₉) / n₆
(l=>)	\| d₁ \| = 2 * t * (n₄ / n₅) * (n₂ / n₃) * (n₆ / (n₆ + n₉))

The same method can be applied to calculate the required size for wheel 14: If rot₁ is held 0, rot₈ has to be exactly one turn when wheel 14 of diameter d₁₄ covered one full circle (this time centre at wheel 1's contact to the ground, radius equal to track width t):

(p:)	d₁₄ * rot₁₄ * PI = 2 * t * PI
(q:)	rot₁ = 0
(j=>)	rot₆ = 0
(a=>)	n₉ * rot₉ + n₆ * 0 - (n₉ + n₆) * 1 = 0
(=>r:)	rot₉ = (n₆ + n₉) / n₉
(k=>)	rot₁₄ = (n₁₀ / n₁₁) * (n₁₂ / n₁₃) * (n₆ + n₉) / n₉
(p=>)	d₁₄ = 2 * t * (n₁₁ / n₁₀) * (n₁₃ / n₁₂) * (n₉ / (n₆ + n₉))

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	n₄ = [teeth]	Spur gear, fixed to wheel 4
5	n₅ = [teeth]	Spur gear, freerunning on central vertical shaft
6	n₆ = [teeth]	Bevel gear, fixed to wheel 5
7	n₇ = [teeth]	Bevel gear, freerunning on short shaft, which in turn is fixed to central vertical shaft
8	-	Platform, fixed to central vertical shaft
9	n₉ = [teeth]	Bevel gear, freerunning on central vertical shaft
10	n₁₀ = [teeth]	Spur gear, fixed to wheel 9
11	n₁₁ = [teeth]	Spur gear, freerunning on vertical shaft
12	n₁₂ = [teeth]	Bevel gear, fixed to wheel 11
13	n₁₃ = [teeth]	Bevel gear, fixed to wheel 14
T	t = [length units]	Track width

Derived Parameters

Reference	Size	Description
1	d₁ = [units of t]	Roadwheel, freerunning on axle
14	d₁₄ = [units of t]	Roadwheel, freerunning on axle