Is my textbook in error about this geometric optics relation?

Given a centered optical system (having an axis of rotational symmetry), let $H_1$ be the intersection of the optical axis and the "principal object plane" (I'm studying in French and have no idea how to translate that... it's the plane whose image is a plane with a scaling factor of 1). Let $H_2$ be the intersection of the optical axis and the "principal focal plane", in other words the image of $H_1$. My book states that the distances between those points and the focal points, $f_1=H_1F_1$ and $f_2=H_2F_2$, verify the following relations:

$$n_1f_1=-n_2f_2$$ $$\frac{n_2}{f_2}=-\frac{n_1}{f_1}=V$$

Where V is the vergence. But these seem to contradict eachother. From relation 1, we have $$n_1=-\frac{n_2f_2}{f_1}$$ Substituting into the second relation: $$\frac{n_2}{f_2}=\frac{n_2f_2}{f_1^2}$$ $$1=\frac{f_2^2}{f_1^2}$$ $$f_1=\pm f_2$$ But the book then goes on to say that that's only true when $n_1=n_2$.

• A diagram will probably help. – Emilio Pisanty May 3 '13 at 14:58
• @EmilioPisanty Added – Jack M May 4 '13 at 8:28