With the above notation, we have

(8.23)			$\displaystyle\deg_{{\mathcal{O}}_{\mathbb{P}(E)}(1)}(\mathbb{P}(E))=\frac{(n+r)!}{n!}\int_{\Delta^{r}}L(y)^{n}\,\text{\rm d}y$
(8.24)			$\displaystyle\operatorname{h}_{\overline{{\mathcal{O}}_{\mathbb{P}(E)}(1)}}(\mathbb{P}(E))=\frac{(n+r+1)!}{(n+1)!}\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})\int_{\Delta^{r}}L(y)^{n+1}\,\text{\rm d}y$
		$\displaystyle\hskip 160.0pt-\frac{(n+r+1)!}{2\,n!}\int_{\Delta^{r}}L(y)^{n}\varepsilon_{r}(y)\,\text{\rm d}y,$

where $\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}}(\mathbb{P}^{n})=\sum_{h=1}^{n}\sum_{j=1}^{h}\frac{1}{2j}$ is the height of the projective space relative to the Fubini-Study metric.