22 RELATIONS PERTAINING SIMPLY [BoOK I. 



If the variations of the anomaly v are wanted in seconds, both parts also of 

 dv must be expressed in this manner, that is, it is necessary to take for Jc the value 

 3548&quot;.1S8 given in article 6. If, moreover, $p = q is introduced instead of p, the 

 formula will have the following form : 



, &amp;lt;qj. 

 dz&amp;gt;z=-* idt --- c 



rr 



in which are to be used the constant logarithms 



log * \l 2 = 3.7005215724, log 3 k \/ } = 3.8766128315. 

 Moreover the differentiation of the equation 



P 



T ^^z 



2cos 2 ^-t 

 furnishes 



= -(- tan i v d v. 



r p 



or by expressing dv by means of d^ and dp, 



d . 

 * 



\p 

 By substituting for t its value in v, the coefficient of dp is changed into 



1 3tan 2 iw ptan^if 1 /i t i . 9 1 ,-21 1-21 9 1 \ 



* _. m___ f I JL. [ _ JL T lll* * 41 - O O1V|* -Jf )) -Jt O1T~l 11 TOYl* * Jt I 



&quot;- -. -. I ff &quot;T&quot; 2 ttlll 5 V ~ -()- bill j V 9 bill 3 V tclll 9 V I 



p irr 4rr r \ 



but the coefficient of d^ becomes - . From this there results 



r \IP 



, . ks\n v , , 



d r = cos t&amp;gt; d jt? -| T d r, 



or if we introduce q for p 



d-, , 

 r::= cos pd- 



The constant logarithm to be used here is log \j J = 8.0850664436. 



21. 



In the HYPERBOLA,9 and E would become imaginary quantities, to avoid 

 which, other auxiliary quantities must be introduced in the place of them. We 

 have already designated by y&amp;gt; the angle of which the cosine =-, and we have 

 found the radius vector 



