SECT. 1.] TO POSITION IN THE ORBIT. 25 



the mass \i (which we can assume to be indeterminable for a body moving in an 

 hyperbola) is neglected, the equation assumes the following form : 



).kt 



VT 



XL -- , 



or by introducing F, 



I e tan F log tan (45 + $ F] = . 



6 



Supposing Brigg s logarithms to be used, we have 



log X = 9.6377843113, log 1 7c = 7.8733657527 ; 



but a little greater precision can be attained by the immediate application of the 

 hyperbolic logarithms. The hyperbolic logarithms of the tangents are found in 

 several collections of tables, in those, for example, which SCHULZE edited, and still 

 more extensively in the Magnus Canon Triangular. Logurtthmicus of BENJAMIN URSIN, 

 Cologne, 1624, in which they proceed by tens of seconds. 



Finally, formula XI. shows that opposite values of t correspond to reciprocal 

 values of u, or opposite values of F and v, on which account equal parts of the 

 hyperbola, at equal distances from the perihelion on both sides, are described in 

 equal times. 



23. 



If we should wish to make use of the auxiliary quantity u for finding the 

 time from the true anomaly, its value is most conveniently determined by means 

 of equation IV. ; afterwards, formula II. gives directly, without a new calculation, 



p by means of r, or r by means of p. Having found u, formula XI. will give the 



ikt 

 quantity =-, which is analogous to the mean anomaly in the ellipse and will be 



5* 

 denoted by N, from which will follow the elapsed time after the perihelion transit. 



Since the first term of N, that is Ji!^I 2 may, by means of formula VIII. be 



made 4-4 - , the double computation of this quantity will answer for testing 

 its accuracy, or, if preferred, JV can be expressed without u, as follows : 



XII. ^V = 



cos 



- ___ 



2 cos ^ (v -f- u&amp;gt;) cos i (v w) cos (v 



4 



