coordinate transormation

Question

In Einstein's "The Meaning of Relativity" I don't understand the relation between $b_να$ in equation (3a) and λ that pops up in equation (2b). I understand the fact that there's a linear transformation between the two coordinate systems and hence the form of (2b) equation. https://en.wikisource.org/wiki/The_Meaning_of_Relativity/Lecture_1

Answer 1

It just expresses the fact that the intervals are constant on their own, so they must be proportional. The statement goes as

\begin{align} \sum \Delta x_\nu^2 &= \text{const} \tag{2} \\ \sum \Delta {x'}_\nu^2 &= \text{const}' \tag{2a} \end{align}

Now write

$$ \text{const}' = \lambda ~\text{const} $$

for some constant $\lambda$. This is always possible to do if both constants are non-zero. Replacing this back into the original equations you get

$$ \underbrace{\sum \Delta {x'}_\nu^2}_{\text{const}} = \lambda \underbrace{\sum \Delta {x}_\nu^2}_{\text{const}'} $$