以下の計算では，既に証明しているつぎの式を用いる： \[ \frac{\overline{ OP'} - r }{ \Delta t} \ \ \ \longrightarrow \ v\ cos( \tau ) \ \ \ ( \Delta t \longrightarrow 0 ) \\ \ \\ \frac{ 1 - cos( a ) }{\Delta t } \ \ \ \longrightarrow \ 0 \ \ \ ( \Delta t \longrightarrow 0 ) \\ \ \\ \frac{ sin( a ) }{\Delta t } \ \ \longrightarrow \ \frac{ v\ sin(\tau) }{r} \ \ \ \ ( \Delta t \longrightarrow 0 ) \] \[ w_x = 0 \\ w_y = r\ \Omega \\ \ \\ w'_x = - \overline{ OP'}\ \Omega\ ( sin(a)\ cos(\Omega \Delta t) + cos(a)\ sin(\Omega \Delta t) \\ w'_y = \overline{ OP'}\ \Omega\ ( cos(a)\ cos(\Omega \Delta t) - sin(a)\ sin(\Omega \Delta t) ) \\ \] \[ \frac{ w'_x - w_x }{ \Delta t } \\ \ \\ = - \overline{ OP'}\ \Omega\ \frac{ 1 }{ \Delta t }\ ( sin(a)\ cos(\Omega \Delta t) + cos(a)\ sin(\Omega \Delta t) ) - 0 ) ) \\ \ \\ = - \overline{ OP'}\ \Omega\ \frac{ 1 }{ \Delta t }\ ( sin(a) - sin( a ) + sin(a)\ cos(\Omega \Delta t) + cos(a)\ sin(\Omega \Delta t) ) \\ \ \\ = - \overline{ OP'}\ \Omega\ ( \frac{ sin(a) }{ \Delta t } - sin( a )\ \frac{ 1 - cos(\Omega \Delta t)}{ \Delta t } + cos(a)\ \frac{ sin(\Omega \Delta t) }{ \Delta t } ) \\ \ \\ = - \overline{ OP'}\ \Omega\ ( \frac{ sin(a) }{ \Delta t } - sin( a )\ ( \Omega^2\ \Delta t )\ \frac{ 1 - cos(\Omega \Delta t)}{ ( \Omega\ \Delta t )^2 } + cos(a)\ \Omega\ \frac{ sin(\Omega \Delta t) }{ \Omega\ \Delta t } ) \\ \ \\ \longrightarrow \ - r\ \Omega\ ( \frac{ v\ sin( \tau ) }{ r }\ - 0 + \Omega ) = - v\ \Omega\ sin( \tau ) - r\ \Omega^2 \ \ \ ( \Delta t \rightarrow 0 ) \\ \] \[ \frac{ w'_y - w_y }{ \Delta t } \\ \ \\ = \frac{ 1 }{ \Delta t }\ ( \overline{ OP'}\ \Omega\ ( cos(a)\ cos(\Omega \Delta t) - sin(a)\ sin(\Omega \Delta t) ) \\ \quad \quad \quad - r\ \Omega ) \\ \ \\ = \frac{ 1 }{ \Delta t }\ ( \overline{ OP'}\ \Omega\ ( cos(a) - cos(a) + cos(a)\ cos(\Omega \Delta t) - sin(a)\ sin(\Omega \Delta t) ) \\ \quad \quad \quad - r\ \Omega ) \\ \ \\ = \frac{ 1 }{ \Delta t }\ ( \overline{ OP'}\ \Omega\ ( cos(a) - cos(a)\ ( 1 - cos(\Omega \Delta t) ) - sin(a)\ sin(\Omega \Delta t) ) \\ \quad \quad \quad - r\ \Omega ) \\ \ \\ = \frac{ 1 }{ \Delta t }\ ( \overline{ OP'}\ \Omega\ ( - cos(a)\ ( 1 - cos(\Omega \Delta t) ) - sin(a)\ sin(\Omega \Delta t) ) \\ \quad \quad \quad + \overline{ OP'}\ \Omega\ cos(a) - r\ \Omega ) \\ \ \\ = \frac{ 1 }{ \Delta t }\ ( \overline{ OP'}\ \Omega\ ( - cos(a)\ ( 1 - cos(\Omega \Delta t) ) - sin(a)\ sin(\Omega \Delta t) ) \\ \quad \quad \quad + \overline{ OP'}\ \Omega\ cos(a) - \overline{ OP'}\ \Omega + \overline{ OP'}\ \Omega - r\ \Omega ) \\ \ \\ = \overline{ OP'}\ \Omega\ ( - cos(a)\ \frac{ 1 - cos(\Omega \Delta t) }{ \Delta t } - sin(a)\ \frac{ sin(\Omega \Delta t) }{ \Delta t }\ )\\ \quad \quad \quad - \overline{ OP'}\ \Omega\ \frac{ 1 - cos(a) }{ \Delta t } + \Omega\ \frac{ \overline{ OP'} - r }{ \Delta t } ) \\ \ \\ \ \\ = \overline{ OP'}\ \Omega\ ( - cos(a)\ ( \Omega^2\ \Delta t )\ \frac{ 1 - cos(\Omega \Delta t) }{ \Delta t } - sin(a)\ \Omega\ \frac{ sin(\Omega \Delta t) }{ \Omega\ \Delta t }\ )\\ \quad \quad \quad - \overline{ OP'}\ \Omega\ \frac{ 1 - cos(a) }{ ( \Omega] \Delta t )^2 } + \Omega\ \frac{ \overline{ OP'} - r }{ \Delta t } ) \\ \ \\ \ \\ \longrightarrow \ r\ \Omega\ ( 0 - 0 ) - 0 + \Omega\ v\ cos( \tau ) = v\ \Omega\ cos( \tau ) \ \ \ ( \Delta t \rightarrow 0 ) \\ \] よって， \[ \frac{ {\bf w'} - {\bf w} }{\Delta t} \longrightarrow ( - v\ \Omega\ sin(\tau) - r\ \Omega^2,\ v\ \Omega\ cos(\tau) ) \ \ \ ( \Delta t \rightarrow 0 ) \]