whence
(8)
| ∂ẋ | = | ∂x | . |
| ∂q̇r | ∂qr |
Also
(9)
| d | ( | ∂x | ) = | ∂2x | q̇1 + | ∂2x | q̇2 + ... + | ∂2x | q̇r = | ∂ẋ | . |
| dt | ∂qr | ∂q1∂qr | ∂q2∂qr | ∂qn∂qr | ∂qr |
Hence
(10)
| ẍ | ∂x | = | d | ( ẋ | ∂x | ) − ẋ | d | ( | ∂x | ) = | d | ( ẋ | ∂ẋ | ) − ẋ | ∂ẋ | . |
| ∂qr | dt | ∂qr | dt | ∂qr | dt | ∂q̇r | ∂qr |
By these and the similar transformations relating to y and z the equation (6) takes the form