I am reading Alan Turing's "The Chemical Basis of Morphogenesis" article, http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf, and I can't understand where the last expression of the article's 69th page comes from. He says that it's the expression of the superficial part of the Laplacian in spherical coordinates. The expression I am referring to is the next one,
\begin{equation*} \frac{1}{\rho^2}\frac{\partial^2V}{\partial\phi^2}+\frac{1}{\rho^2\sin^2{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta\frac{\partial V}{\partial\theta}}\right). \end{equation*}
I have compared it with the next usual expression of the Laplacian in spherical coordinates,
\begin{equation*} \frac{1}{\rho^2}\frac{\partial}{\partial\rho}\left(\rho^2\frac{\partial V}{\partial\rho}\right)+\frac{1}{\rho^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial V}{\partial\theta}\right)+\frac{1}{\rho^2\sin^2{\theta}}\frac{\partial^2 V}{\partial\phi^2}. \end{equation*}
I guess that when Alan Turing says "the superficial part of the Laplacian" he is referring to the surface of the sphere, so, in that case $\rho$ could be constant and the part $\frac{1}{\rho^2}\frac{\partial}{\partial\rho}\left(\rho^2\frac{\partial V}{\partial\rho}\right)$ of the usual expression of the Laplacian in spherical coordinates disappears. But what happens with the next part? \begin{equation*} \frac{1}{\rho^2\sin{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta}\frac{\partial V}{\partial\theta}\right)+\frac{1}{\rho^2\sin^2{\theta}}\frac{\partial^2 V}{\partial\phi^2}. \end{equation*}
It's not the same as what Turing presents,
\begin{equation*} \frac{1}{\rho^2}\frac{\partial^2V}{\partial\phi^2}+\frac{1}{\rho^2\sin^2{\theta}}\frac{\partial}{\partial\theta}\left(\sin{\theta\frac{\partial V}{\partial\theta}}\right). \end{equation*}
This question is driving me crazy because I have tried to think about it and I have also tryed to search for things that could be related to this, but I couldn't find the answer.
I would be very happy if I could discover what the expression that Alan Turing presents in his article means. Does anyone know something about this topic, please?
Thank you very much for your help.
Edit: It seems that the superficial part of the Laplacian is not the restriction of the Laplacian to the surface of the sphere:
How to define Surface Laplacian on the sphere with radius 1.