104( Prof. Encke 07i the Calculation 



(i?) a' sinM sin(U + aj')= v sinV; a' sin «' sin(U' + oJ) = v' sin V 5 

 i' sin Mcos(U + w') = v [cosV ; V sin u'cos(U' + w') = v' cosV ; 

 — sin ^'. V sin V = A ; — sin 0'. v' sin V = a'. 



and the calculation of the place is then contained in the fol- 

 lowing formulae : 



E'- sin$'sinE' = (^-T)ju.' 



(4-8) qsmp = v' sin (E' + V') + x' 



q cosp = V sin (E' + V) + X. 



The constants thus determined maj', if the elements have 

 been previously calculated, be more easily found by these 

 equations : 



, . Qv V sin V = —I cos P ; v ' sin V = — Z sin P 

 *^ ' ' V cos V = — /' sin Q ; v' cos Y' = + I' cos Q 



and vice versa, the latter may be found from the former. It 



may, however, be interesting to know the times in which the 



maxima and minima of visible distance and angular velo- 



cl J) 

 city take place. The quantity g^ , = k being constant, the 



smallest and greatest distances will coincide, as in the true 

 ellipsis, with the greatest and smallest angular velocity. Dif- 

 ferentiating the formulas (42) or (48) with regular to E', we 

 obtain, 



2§ ^f — {v'' cos 2V' + V- cos 2V) sin 2E' 



+ (/ - sin 2V' + v- sin 2V) cos 2E' 

 + 2(/2 sin V'- + v"- sin V^) sin ^' sin E' 

 — (v'^ sin 2V' + v^ sin 2V) sin f>Vcos E'. 

 The calculation will be simplified by introducing the ele- 

 ments by means of (49) and (46). As 



v"' cos 2V' + v- cos 2V = — (a-— Z>^) cos 2F 

 v'- sin 2V' + v' sin 2V =-(a*-i-) sin 2F 

 2(/^ sin V' + v^ sin V^) = 2/^ 

 The above equation may be written thus : 



2g -^ = - {a^-h") sin 2{YI + Y) + 2P sin <f>' sin E' 

 ^^ + {a^-b") sin f' sin 2F cos E'. 



■p 

 Substituting for sin (p' its equivalent -r- and for 2Z* the sum 



of the two last expressions of (46), it becomes : 



2q A, = _(g2_^3) sin 2(E' + F) 



oR 



+ -J- {a^ cos F sin (E' + Y) -b^ sin F cos (E' + F) } 



Supposing, 



