Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/56

 previous injunction, that, if the variations of the angles $v$ and $\psi$  are conceived to be expressed, not in parts of the radius, but in seconds, either all the terms containing $\mathrm{d} v, \mathrm{~d} \psi,$  must be divided by 206264.8, or all the remaining terms must be multiplied by this number.

29.
Since the auxiliary quantities $\varphi, E, M,$ employed in the ellipse obtain imaginary values in the hyperbola, it will not be out of place to investigate their connection with the real quantities of which we have made use: we add therefore the principal relations, in which we denote by $i$  the imaginary quantity $\sqrt{ }-1.$

$$\begin{aligned} & \sin \varphi=e=\frac{1}{\cos \psi} \\ & \tan \left(45^{\circ}-\frac{1}{2} \varphi\right)=\sqrt{\frac{1-e}{1+e}}=i \sqrt{\frac{e-1}{e+1}}=i \tan \frac{1}{2} \psi \\ & \tan \varphi=\frac{1}{2} \operatorname{cotan}\left(45^{\circ}-\frac{1}{2} \varphi\right)-\frac{1}{2} \tan \left(45^{\circ}-\frac{1}{2} \varphi\right)=-\frac{i}{\sin \psi} \\ & \cos \varphi=i \tan \psi \\ & \varphi=90^{\circ}+i \log (\sin \varphi+i \cos \varphi)=90^{\circ}-i \log \tan \left(45^{\circ}+\frac{1}{2} \psi\right) \\ & \tan \frac{1}{2} E=i \tan \frac{1}{2} F=\frac{i(u-1)}{u+1} \\ & \frac{1}{\sin E}=\frac{1}{2} \operatorname{cotan} \frac{1}{2} E+\frac{1}{2} \tan \frac{1}{2} E=-i \operatorname{cotan} F, \end{aligned}$$

or

$$\begin{aligned} & \sin E=i \tan F=\frac{i(u u-1)}{2 u} \\ & \operatorname{cotan} E=\frac{1}{2} \operatorname{cotan} \frac{1}{2} E-\frac{1}{2} \tan \frac{1}{2} E=-\frac{i}{\sin F} \end{aligned}$$

or

$$\begin{aligned} & \tan E=i \sin F=\frac{i(u u-1)}{u u+1} \\ & \cos E=\frac{1}{\cos F}=\frac{u u+1}{2 u} \\ & i E=\log (\cos E+i \sin E)=\log \frac{1}{u}, \end{aligned}$$

or

$$\begin{aligned} & E=i \log u=i \log \left(45^{\circ}+\frac{1}{2} F\right) \\ & M=E-e \sin E=i \log u-\frac{i e(u u-1)}{2 u}=-\frac{i N}{\lambda} \end{aligned}$$

The logarithms in these formulas are hyperbolic.