58 RELATIONS PERTAINING SIMPLY [BoOK I. 



The following formulas are easily deduced from the combination of the pre 

 ceding : 



V. sin (u X -{- 8 ) = 2 sin 2 i sin u cos (X 8 ) 

 VI. sin (u X -f- 8 ) = tan J z sin ft cos (X 8 ) 

 VII. sin ( X -(- 8 ) = tan i z tan ft cos w 

 VIII. sin (u -\- X 8 ) = 2 cos 2 J j sin cos (X 8 ) 

 IX. sin (u -4- X 8 ) = cotan i sin ft cos (X 8 ) 



X. sin (w -{- X 8 ) = cotan a tan ft cos w. 



The angle u X -4- 8, when a is less than 90, or w -|- X 8, when i is more 

 than 90, called, according to common usage, the reduction to the ecliptic, is, in fact, 

 the difference between the heliocentric longitude X and the longitude in orbit, 

 which last is by the former usage 8 + , by ours 8 -)- u. When the inclination 

 is small or differs but little from 180, the same reduction may be regarded as a 



^ 



quantity of the second order, and in this case it will be better to compute first ft 

 by the formula III., and afterwards X by VII. or X., by which means a greater 

 precision will be attained than by formula I. 



If a perpendicular is let fall from the place of the heavenly body in space 

 upon the plane of the ecliptic, the distance of the point of intersection from the 

 sun is called the curtate distance. Designating this by /, the radius vector likewise 

 by r, we shall have 



XI. / = r cos ft. 



51. 



As an example, we will continue further the calculations commenced in arti 

 cles 13 and 14, the numbers of which the planet Juno furnished. We had 

 found above, the true anomaly 3151 23&quot;.02, the logarithm of the radius vector 

 0.3259877: now let i == 136 44&quot;.10, the distance of the perihelion from the 

 node = 24110 20&quot;.57, and consequently u = 19611 43&quot;.59 j finally let 8 = 

 171 7 48&quot;.73. Hence we have : - 

 log tan u .... 9.4630573 log sin (X 8). . . . 9.4348691 



log cos i .... 9.9885266 log tan i 9.3672305 



log tan (X ) .. 9.4515839 log tan ft ...... 8.8020996 



