Given any \(x\),
\(\cfrac { \partial x\, }{ \partial x } =1\)
\(\cfrac { \partial ^{ 2 }\, x }{ \partial \, x^{ 2 } } =0=\cfrac { \partial ^{ 2 }\, x }{ \partial \, x\partial t } \cfrac { \partial t\, }{ \partial \, x } =\cfrac { \partial ^{ 2 }\, x }{ \partial t\partial t } \cfrac { \partial t\, }{ \partial \, x } \cfrac { \partial t\, }{ \partial \, x } =\cfrac { \partial ^{ 2 }\, x }{ \partial t^{ 2 } } \cfrac { \partial t\, }{ \partial \, x } \cfrac { \partial t\, }{ \partial \, x } \)
So for all \(x\),
\(\cfrac { \partial ^{ 2 }\, x }{ \partial t^{ 2 } } \cfrac { \partial t\, }{ \partial \, x } \cfrac { \partial t\, }{ \partial \, x } =0\)
ie,
\(\cfrac{a}{v^2}=0\)
In fact,
\(\cfrac { \partial ^{ 2 }x\quad }{ \partial \, x^{ 2 } } =0\)
\(=\cfrac { \partial \, \quad }{ \partial \, x } \{ \cfrac { \partial x\, }{ \partial \, x } \} \)
\(=\cfrac { \partial \, \quad }{ \partial \, t } \{ \cfrac { \partial x\, }{ \partial \, x } \} \cfrac { \partial t\, }{ \partial \, x } \)
\(=\cfrac { \partial \, \quad }{ \partial \, t } \{ \cfrac { \partial x\, }{ \partial \, t } \cfrac { \partial t\, }{ \partial \, x } \} \cfrac { \partial t\, }{ \partial \, x } \)
\(=\cfrac { \partial \, \quad }{ \partial \, t } \{ \cfrac { \partial x\, }{ \partial \, t } \} \cfrac { \partial t\, }{ \partial \, x } +\cfrac { \partial \, \quad }{ \partial \, t } \{ \cfrac { \partial t\, }{ \partial \, x } \} \cfrac { \partial x\, }{ \partial \, t } \)
\(= \cfrac { \partial ^{ 2 }x\, \quad }{ \partial \, t^{ 2 } } \cfrac { \partial t\, }{ \partial \, x } -\cfrac { 1 }{ \{ \cfrac { \partial x\, }{ \partial \, t } \} ^{ 2 } } \cfrac { \partial \, \quad }{ \partial \, t } \{ \cfrac { \partial x\, }{ \partial \, t } \} \cfrac { \partial x\, }{ \partial \, t } \)
\(=\cfrac { \partial ^{ 2 }x\, \quad }{ \partial \, t^{ 2 } } \cfrac { \partial t\, }{ \partial \, x } -\cfrac { 1 }{ \{ \cfrac { \partial x\, }{ \partial \, t } \} } \cfrac { \partial ^{ 2 }x\, \quad }{ \partial \, t^{ 2 } } \)
\(=\cfrac { \partial ^{ 2 }x\, \quad }{ \partial \, t^{ 2 } } \cfrac { \partial t\, }{ \partial \, x } -\cfrac { \partial ^{ 2 }x\, \quad }{ \partial \, t^{ 2 } } \cfrac { \partial t }{ \partial x }=0 \)
non sequitur.
