SECT. 1.] 



TO POSITION IN THE ORBIT. 



27 



will come out greater in the ratio 1 : X, than if Brigg s logarithms were used. 

 Our example treated according to this method is as follows : 



log tan 4 y .... 9.5318179 

 log tan 4 v . 9.2201009 



log tan 4 F 



8.7519188 



log e . . . . . . . 0.1010188 

 log tan I 7 9.0543366 



9.1553554 



etznF= 0.14300638 



hyp. log tan (45 + 4 F}= 0.11308666 



N= 0.02991972 



log& ...... 8.2355814) 



| log b 0.9030900 / 



41 7 =313 58&quot;.12 



C. hyp. log cos 4 (v 1/&amp;gt;) = 0.01342266 

 C. hyp. log cos 4 (v + Y) = 0.12650930 



Difference 



= 0.11308664 



log^V 8.4759575 



Difference 7.3324914 



logl 1.1434661 



t= 13.91445 



25. 



For the solution of the inverse problem, that of determining the true anomaly 

 and the radius vector from the time, the auxiliary quantity u or F must be first 

 derived from N= &quot;kk b ^t by means of equation XI. The solution of this tran 

 scendental equation will be performed by trial, and can be shortened by devices 

 analogous to those we have described in article 11. But we suffer these to pass 

 without further explanation ; for it does not seem worth while to elaborate as 

 carefully the precepts for the hyperbolic motion, very rarely perhaps to be exhib 

 ited in celestial space, as for the elliptic motion, and besides, all cases that can 

 possibly occur may be solved by another method to be given below. After 

 wards F or u will be found, thence v by formula III., and subsequently r will be 

 determined either by II. or VIII. ; v and r are still more conveniently obtained 

 by means of formulas VI. and VII. ; some one of the remaining formulas can be 

 called into use at pleasure, for verifying the calculation. 



