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/43

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15.

Since, as we have seen, the mean anomaly ${\textstyle {\dot {M}}}$ is completely determined by means of ${\textstyle v}$ and ${\textstyle \varphi ,}$ in the same manner as ${\textstyle v}$ by ${\textstyle M}$ and ${\textstyle \varphi ,}$ it is evident, that if all these quantities are regarded as variable together, an equation of condition ought to exist between their differential variations, the investigation of which will not be superfluous. By differentiating first, equation VII., article 8, we obtain

$${\frac {dE}{\sin E}}={\frac {dv}{\sin v}}-{\frac {d\varphi }{\cos \varphi }}$$

by differentiating likewise equation XII., it becomes

$$\mathrm {d} M=(1-e\cos E)\mathrm {d} E-\sin E\cos \varphi \mathrm {d} \varphi .$$

If we eliminate ${\textstyle \mathrm {d} E}$ from these differential equations we have

$$\mathrm {d} M={\frac {\sin E(1-e\cos E)}{\sin v}}\mathrm {~d} v-\left(\sin E\cos \varphi +{\frac {\sin E(1-e\cos E)}{\cos \varphi }}\right)\mathrm {d} \varphi$$

or by substituting for ${\textstyle \sin E,1-e\cos E,}$ their values from equations VIII., III.,

$$\mathrm {d} M={\frac {rr}{aa\cos \varphi }}\mathrm {d} v-{\frac {r(r+p)\sin v}{aa\cos ^{2}\varphi }}\mathrm {d} \varphi$$

or lastly, if we express both coefficients by means of ${\textstyle v}$ and ${\textstyle \varphi }$ only,

$$\mathrm {d} M={\frac {\cos ^{3}\varphi }{(1+e\cos v)^{2}}}\mathrm {~d} v-{\frac {(2+e\cos v)\sin v\cos ^{2}\varphi }{(1+e\cos v)^{2}}}\mathrm {~d} \varphi .$$

Inversely, if we consider ${\textstyle v}$ as a function of the quantities ${\textstyle M,\varphi ,}$ the equation has this form:-

$$\mathrm {d} v={\frac {aa\cos \varphi }{rr}}\mathrm {~d} M+{\frac {(2+e\cos v)\sin v}{\cos \varphi }}\mathrm {d} \varphi$$

or by introducing ${\textstyle E}$ instead of ${\textstyle v}$

$$\mathrm {d} v={\frac {aa\cos \varphi }{rr}}\mathrm {~d} M+{\frac {aa}{rr}}(2-e\cos E-ee)\sin E\mathrm {~d} \varphi .$$