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/50

In the same manner, by adding 1 to both sides, it becomes

VII. ${\textstyle \cos {\frac {1}{2}}v{\sqrt {}}r=\cos {\frac {1}{2}}F{\sqrt {\frac {p}{(e+1)\cos F}}}=\cos {\frac {1}{3}}F{\sqrt {\frac {(e-1)b}{\cos F}}}}$

$$={\frac {1}{2}}(u+1){\sqrt {\frac {p}{(e+1)u}}}={\frac {1}{2}}(u+1){\sqrt {{\frac {(e-1)b}{u}}.}}$$

By dividing VI. by VII. we should reproduce III.: the multiplication produces VIII. ${\textstyle r\sin v=p\operatorname {cotan} \psi \tan F=b\tan \psi \tan F}$

$$={\frac {1}{2}}p\operatorname {cotan} \psi \left(u-{\frac {1}{u}}\right)={\frac {1}{2}}b\tan \psi \left(u-{\frac {1}{u}}\right){\text{. }}$$

From the combination of the equations II. V. are easily derived

IX. ${\textstyle r\cos v=b\left(e-{\frac {1}{\cos F}}\right)={\frac {1}{2}}b\left(2e-u-{\frac {1}{u}}\right),}$ X. ${\textstyle r=b\left({\frac {e}{\cos F}}-1\right)={\frac {1}{2}}b\left(e\left(u+{\frac {1}{u}}\right)-2\right).}$

22.

By the differentiation of the formula IV. (regarding ${\textstyle \psi }$ as a constant quantity) we get

$${\frac {\mathrm {d} u}{u}}={\frac {1}{2}}\left(\tan {\frac {1}{2}}(v+\psi )-\tan {\frac {1}{2}}(v-\psi )\right)\mathrm {d} v={\frac {r\tan \psi }{p}}\mathrm {~d} v$$

hence,

$$rr\mathrm {~d} v={\frac {pr}{u\tan \psi }}\mathrm {d} u$$

or by substituting for ${\textstyle r}$ the value taken from ${\textstyle \mathrm {X} .}$

$$rr\mathrm {~d} v=bb\tan \psi \left({\frac {1}{2}}e\left(1+{\frac {1}{uu}}\right)-{\frac {1}{u}}\right)\mathrm {d} u$$

Afterwards by integrating in such a manner that the integral may vanish at the perihelion, it becomes

$$\int rr\mathrm {d} v=bba\tan \psi \left({\frac {1}{2}}e\left(u-{\frac {1}{u}}\right)-\log u\right)=kt{\sqrt {p}}{\sqrt {\left(1+\mu \right)}}=kt\tan \psi {\sqrt {b}}{\sqrt {1+\mu }}$$

The logarithm here is the hyperbolic; if we wish to use the logarithm from Brigg’s system, or in general from the system of which the modulus ${\textstyle =\lambda ,}$ and