The physics of time-space parallels that of space-time. Here is a derivation of the corresponding equations of motion, paralleling the exposition at the *Physics Hypertextbook*.

The one-dimensional equations of motion for constant modulation:

Let *t* = time, *t*_{0} = initial time, *r* = displacement, u = legerity, *u*_{0} = initial legerity, *b* = modulation.

First equation of motion

*b* = d*u* / d*r*

d*u* = *b* d*r*

∫ d*u* from *u*_{0} to *u* = ∫ *b* d*r* from 0 to *r*

*u* – *u*_{0} = *br*

*u* = *u*_{0} + *br*

Second equation of motion

*u* = d*t* / d*r*

d*t* = *u* d*r* = (*u*_{0} + *br*) d*r*

∫ d*t* from *t*_{0} to *t* = ∫ (*u*_{0} + *br*) d*r* from 0 to *r*

*t* – *t*_{0} = *u*_{0}*r* + ½*br*²

*t* = *t*_{0} + *u*_{0}*r* + ½*br*²

Third equation of motion

d*u* / d*t* = (d*u* / d*t*) (d*r* / d*r*) = (d*u* / d*r*) (d*r* / d*t*) = *b* / *u*

*u* d*u* = *b* d*t*

∫ *u* d*u* from *u*_{0} to *u* = ∫ *b* d*t* from *t*_{0} to *t*

½(*u*² – *u*_{0}²) = *b*(*t* – *t*_{0})

*u*² = *u*_{0}² + 2*b*(*t* – *t*_{0})

Since displacement is in the denominator and time is in the numerator of legerity and modulation, these equations may be generalized to three dimensions of time.