I am reading L.E. Scriven's "On the dynamics of phase growth", here is the link https://www.sciencedirect.com/science/article/pii/0009250959800191.
My questions are about equation (36) - (42): $$\dot{t}=K(t_{rr}+2r^{-1}t_{r})-r^{-2}R^{2}\dot{R}t_{r},\qquad\quad(36)$$ $$t(r,0)=t(\infty,\theta)=0\qquad\qquad\qquad(37)$$ $$t(R,\theta)=-\tau\qquad\qquad\qquad\qquad(38)$$ $$t_{r}(R,\theta)=K^{-1}\dot{R}(\xi+\omega v\tau).\qquad\qquad(39)$$
then the author said "On dimensional grounds we assume as a solution $t(r,\theta)=t(s)$ and $$r=2\beta{\sqrt{K\theta}}\quad\quad (40)$$ where $$s=r/2{\sqrt{K\theta}}\quad\quad (41).$$, then the equation (36) becomes $$t_{s s}=2t_{s}(-s-s^{-1}+\epsilon\beta^{3}s^{-2})$$
I don't understand where (40) and (41) come from, why we can assume that? And how to get the last equation? Here $t$ is the temperature and $\theta$ is time variable.$R(\theta)$ is a function of time.