Remark

Formulas (6) remain true for $t = \infty$ provided one replaces the rational functions occurring on the right-hand side by their canonical extensions to the projective field $\tilde{A}$ (Chapter II, 2nd ed., App. III, No. 5); in fact, if $t = \infty$, $\varphi$ is the class of $\omega$, and one has $d(\varphi) = -1, s(d(\varphi)) = 0$ and $c(d(\varphi)) = -1$; these are indeed the values taken by the canonical extensions of the right-hand sides for $t = \infty$. The same is true for (7) when $t(\theta) = \infty$, and for (8) when $t(\varphi) = \infty$ or when $1 + \delta t(\varphi)^2 = 0$. Analogously, formula (2) remains true when only one of the elements $t(\varphi), t(\varphi'), t(\varphi)$, for example, is infinite, provided one considers its right-hand side as a rational function of $t(\varphi)$ alone: in fact the product of the similarities $1 + t(\varphi')\omega$ and $\omega$ is $\delta t(\varphi') + \omega$, whereas the value taken by the canonical extension of the right-hand side of (2) is $1/\delta t(\varphi')$. Finally, when $t(\varphi)$ and $t(\varphi')$ are finite and one has $1 + \delta t(\varphi)t(\varphi') = 0$, one has $t(\varphi) + t(\varphi') = 0$ (otherwise $t(\varphi)^2$ would be equal to $1/\delta$, which is impossible (prop. 4)); one may therefore agree that the value of the right-hand side of (2) is $\infty$, and this value is indeed that of the left-hand side. When $t(\varphi)$ and $t(\varphi')$ are both infinite, the right-hand side of (2) is not defined.