Rewriting a system of two equations found in The Chemical Basis of Morphogenesis (A. Turing, 1952)

Question

In reading The Chemical Basis of Morphogenesis by A. Turing, I am unable to follow a small section of his working. On page $47$, Turing states that

\begin{align*} x_r&=\sum_{s=0}^{N-1} \exp\left(\frac{2\pi irs}{N}\right)\xi_s, \\ y_r&=\sum_{s=0}^{N-1} \exp\left(\frac{2\pi irs}{N}\right)\eta_s, \end{align*}

can be written as

\begin{align*} \xi_r&=\frac{1}{N}\sum_{s=1}^{N} \exp\left(-\frac{2\pi irs}{N}\right)x_s, \\ \eta_r&=\frac{1}{N}\sum_{s=1}^{N} \exp\left(-\frac{2\pi irs}{N}\right)y_s. \end{align*}

This does not seem obvious to me. It is given that this can be shown by using the equations \begin{align} \sum_{s=1}^N \exp\left(\frac{2\pi irs}{N}\right)&=0 \ \ \text{if} \ \ 0<r<N, \\ &=N \ \ \text{if} \ \ r=0 \ \ \text{or} \ \ r=N. \end{align} However, I do not see how this helps. I hint would be most helpful.

Answer 1

Put the sums for $x_s$ and $y_s$ in the sums for $\xi$ and $\eta$ and then reverse the order of summation.

Use those last sums to make almost everything go to zero except the desired terms.