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/45

 which, nevertheless, is not a maximum here, since the difference between $v$ and $E$  may still increase beyond $\varphi.$  This last difference becomes a maximum for $\mathrm{d}(v-E)=0$  or for $\mathrm{d} v=\mathrm{d} E,$  where the eccentricity is clearly to be regarded as constant. With this assumption, since in general

$$\frac{\mathrm{d} v}{\sin v}=\frac{\mathrm{d} E}{\sin E}$$

it is evident that we should have $\sin v=\sin E$ at that point where the difference between $v$  and $E$  is a maximum; whence we have by equations VIII., III.,

$$r=a \cos \varphi, e \cos E=1-\cos \varphi, \text { or } \cos E=+\tan \frac{1}{2} \varphi$$

In like manner $\cos v=-\tan \frac{1}{2} \varphi$ is found, for which reason it will follow * that

hence again

$$v=90^{\circ}+\arcsin \tan \frac{1}{2} \varphi, E=90^{\circ}-\arcsin \tan \frac{1}{3} \varphi$$

$$\sin E=V\left(1-\tan ^{2} \frac{1}{2} \varphi\right)=\frac{V \cos \varphi}{\cos \frac{1}{2} \varphi}$$

so that the whole equation of the centre at this point becomes

$$2 \arcsin \tan \frac{1}{2} \varphi+2 \sin \frac{1}{2} \varphi \sqrt{\cos \varphi},$$

the second term being expressed in degrees, etc. At that point, finally, where the whole equation of the centre is a maximum, we must have $\mathrm{d} v=\mathrm{d} M,$ and so according to article $15, r=a \vee \cos \varphi;$  hence we have

$$\cos v=-\frac{1-\cos ^{\frac{3}{2} \varphi} \varphi}{e}, \cos E=\frac{1-\sqrt{\cos \varphi}}{e}=\frac{1-\cos \varphi}{e(1+\sqrt{\cos \varphi)}}=\frac{\tan \frac{1}{2} \varphi}{1+\sqrt{\cos \varphi}},$$

by which formula $E$ can be determined with the greatest accuracy. $E$ being found, we shall have, by equations X., XII.,

$$\text { equation of the centre }=2 \arcsin \frac{\sin \frac{1}{2} \varphi \sin E}{\sqrt[4]{\cos \varphi}}+e \sin E \text {. }$$

We do not delay here for an expression of the greatest equation of the centre by means of a series proceeding according to the powers of the eccentricities, which several authors have given. As an example, we annex a view of the three maxima which we have been considering, for Juno, of which the eccentricity, according to the latest elements, is assumed $=0.2554996.$ ? =0

??=0