# Complex potential in E227?

I have a reason to believe that Euler introduced the complex potential in his Continuation des recherches sur la theorie du mouvement does fluides, published in 1757. However, I am having hard time recognizing where it appears. The relevant place seems to be here:

After establishing that $$(u-iv)(dx+idy)$$ and $$(u+iv)(dx-idy)$$ are integrable, he introduces functions $$\phi$$ and $$\psi$$. The problem is I am not sure what $$\phi:(x+yi)$$ means. Normally, I would expect the expression to involve some kind of derivative, such as $$(u-iv)(dx+idy) = d\phi + id\psi.$$

I am hoping somebody could explain what is going on here.

If you need, Euler continues on the following page as:

Thanks!