One such rule would be \[ \left. \begin{alignedat}{3} R &= \frac{\dd r}{\dd x}\, X &&+ \frac{\dd r}{\dd y}\, Y &&+ \frac{\dd r}{\dd z}\, Z \\ \Theta &= \frac{\dd \theta}{\dd x}\, X &&+ \frac{\dd \theta}{\dd y}\, Y &&+ \frac{\dd \theta}{\dd z}\, Z \\ \Phi &= \frac{\dd \phi}{\dd x}\, X &&+ \frac{\dd \phi}{\dd y}\, Y &&+ \frac{\dd \phi}{\dd z}\, Z \end{alignedat}\right\} \Tag{(20.1)} \] Applying the same rule to the transformation from $(r, \theta, \phi)$ to $(\lambda, \mu, \nu)$ we have \[ \Lambda = \frac{\dd \lambda}{\dd r}\, R + \frac{\dd \lambda}{\dd \theta}\, \Theta + \frac{\dd \lambda}{\dd \phi}\, \Phi, \Tag{(20.2)} \] whence, substituting for $R$, $\Theta$, $\Phi$ from~\Eq{(20.1)} and collecting terms, \begin{align*} \Lambda &= \left(\frac{\dd \lambda}{\dd r}\, \frac{\dd r}{\dd x} + \frac{\dd \lambda}{\dd \theta}\, \frac{\dd \theta}{\dd x} + \frac{\dd \lambda}{\dd \phi}\, \frac{\dd \phi}{\dd x}\right) X + \left(\frac{\dd \lambda}{\dd r}\, \frac{\dd r}{\dd y} + \frac{\dd \lambda}{\dd \theta}\, \frac{\dd \theta}{\dd y} + \frac{\dd \lambda}{\dd \phi}\, \frac{\dd \phi}{\dd y}\right) Y \\ &\qquad\qquad+ \left(\frac{\dd \lambda}{\dd r}\, \frac{\dd r}{\dd z} + \frac{\dd \lambda}{\dd \theta}\, \frac{\dd \theta}{\dd z} + \frac{\dd \lambda}{\dd \phi}\, \frac{\dd \phi}{\dd z}\right) Z \\ &= \frac{\dd\lambda}{\dd x}\, X + \frac{\dd\lambda}{\dd y}\, Y + \frac{\dd\lambda}{\dd z}\, Z, \Tag{(20.3)} \end{align*} which is the same formula as we should have obtained by applying the rule to the direct transformation from $(x, y, z)$ to $(\lambda, \mu, \nu)$. The rule is thus self-consistent. But this is a happy accident, pertaining to this particular rule, and depending on the formula \[ \frac{\dd\lambda}{\dd x} = \frac{\dd \lambda}{\dd r}\, \frac{\dd r}{\dd x} + \frac{\dd \lambda}{\dd \theta}\, \frac{\dd \theta}{\dd x} + \frac{\dd \lambda}{\dd \phi}\, \frac{\dd \phi}{\dd x}, \] and amid the apparently infinite choice of formulae it will not be easy to find others which have this self-consistency. The above rule is that already given for the contravariant vector~\Eq{(19.1)}. The rule for the covariant vector is also self-consistent. There do not appear to be any other self-consistent rules for the transformation of a set of three numbers (or four numbers for four coordinates)\footnotemark.\footnotetext {Except that we may in addition multiply by any power of the Jacobian of the transformation. This is self-consistent because \[ \frac{\dd(x, y, z)}{\dd(r, \theta, \phi)} · \frac{\dd(r, \theta, \phi)}{\dd(\lambda, \mu, \nu)} = \frac{\dd(x, y, z)}{\dd(\lambda, \mu, \nu)}. \] Sets of numbers transformed with this additional multiplication are degenerate cases of tensors of higher rank considered later. See \SecRefs{48}, \SecNum{49}.}