Remark

When –1 belongs to the class mod. $(A^*)^2$ in question, one generally chooses $\omega$ in such a way that $\omega^2 = -1$, which determines it up to sign. When this class contains 1 but not –1, one generally chooses $\omega$ in such a way that $\omega^2 = 1$, which again determines it up to sign.

No. 2

ANGLES

This being so, every element $\varphi$ of $S^+$ can be written, in one and only one way, in the form

(1)

where $c_w(\varphi), s_w(\varphi)$ belong to $A$; the element $s_w(\varphi)/c_w(\varphi)$ of the projective field $\tilde{A}$ (Chap. II, 2nd ed., App. III, No. 5) is denoted by $t_w(\varphi)$; it depends only on the class of $\varphi$ mod. $H$; thus $t_w$ defines, by passing to the quotient, a mapping of $S^+/H$ into the projective field $\tilde{A}$, which we shall again denote by $t_w$ by abuse of language. We shall often write $c, s$ and $t$ instead of $c_w, s_w$ and $t_w$. We have $c_{-w} = c_w, s_{-w} = -s_w$ and $t_{-w} = -t_w$.