Transforming a set of equations from one set of variables to another? Assume $\theta_1,\dots,\theta_n$ are $n$ positive numbers such that $$\theta_1+\dots+\theta_n=1$$ Define $y_{ij}=\frac{\theta_i}{\theta_j}$ for all $i,j \in \{1,\dots,n\}$. Is there a way to transform the above equation in terms of $y_{ij}$. 

No. If you multiply all $\theta_i$ by the same positive number $c\neq 1$, then all $y_{ij}$ stay the same, but $\theta_1+\dots+\theta_n$ changes.
In more details: assume that you have a condition or a set of conditions on $y_{ij}=\frac{\theta_i}{\theta_j}$ which is satisfied if and only if $\theta_1+\dots+\theta_n=1$. Let $\overline{\theta}_i$ be positive numbers such that $\overline{\theta}_1+\dots+\overline{\theta}_n=1$. Then, by assumption, the numbers $\overline{y}_{ij}=\frac{\overline{\theta}_i}{\overline{\theta}_j}$ satisfy that condition(s). Now choose a positive number $c\neq 1$ and set $\theta'_i = c\overline{\theta}_i$. Then $y'_{ij} = \frac{\theta'_i}{\theta'_j} = \overline{y}_{ij}$, so the numbers $y'_{ij}$ satisfy the condition(s) as well, but $\theta'_1+\dots+\theta'_n = c(\overline{\theta}_1+\dots+\overline{\theta}_n) = c\neq 1$.
