SECT. 3.] PLACES IN ORBIT. 143 



and, in the same manner, the interval of time from the perihelion to the time cor 

 responding to the second place, 



^ /2 cos (.AT n) sin (N-\- n) , , / A e-o i 7ir\ , i AS.O i \\ 



t (~ -Wofe^T J - hyP-log tan (45 + JV) tan (45 +)). 



t 



If, therefore, the first time is put = 2 1 i tf, and, therefore, the second = T-\- J t, 

 we have 



whence the tune of perihelion passage will be known ; finally, 



a 



,n j 2 a 2 /etan2n T /^co i \\ 



[24] t = T (^ - log tan (4o + )) , 



which equation, if it is thought proper, can be applied to the final proof of the 

 calculation. 



105. 



To illustrate these precepts, we will make an example from the two places 

 in articles 23, 24, 25, 46, computed for the same hyperbolic elements. Let, 

 accordingly, 



t /_ z , = 4812 / 0&quot;, or/ = 24 6 0&quot;, log r 0.0333585, log/ = 0.2008541, 

 t = 51.49788 days. 



Hence is found 



w = 2 45 28 / .47, I = 0.05796039, 



j^P or the approximate value of h = 0.0644371 ; hence, by table H., 

 \Q%yy 0.0560848, m = 0.05047454, z = 0.00748585, 



& / 



to which in table HE. corresponds C = 0.0000032. Hence the corrected value of 

 h is 0.06443691, 



losyy = 0.0560846, - m = 0.05047456, z= 0.00748583, 

 yy 



which values require no further correction, because f is not changed by them. 

 The computation of the elements is as follows : 



