408 Prof. Encke on the Calculation 



(13) tang2? tangy-tang 2 ?! tang |3-|-tang 2 ? 2 tang a =0 



tang 2 ? tang 2?! tang 2 ? 2 = Q 



sin 2 y sin 2 /3 sin 2 a 



the third of which is the sum of the two others. 



Deriving the value of a b from (10), we shall find many 

 transformations of the same by applying the following three 

 equations, which will easily be found by the above-given re- 

 lations. The three equations are : 



sin(/3 ) sin(/3 + a) = % tang 2 ? sin 2 a sin 2 /3 



(14) sin (y a) sin (y + a) = \ tang 2 ? t sin 2 a sin 2 y 

 sin (y /3) sin (y + /3) = tang 2 ? 2 sin 2 /3 sin 2y 



We obtain accordingly, 



4 sin (y a) sin (y-fa) sin 2 /3 

 (1234) 



(15) 



All which expressions will be applied below. 



As soon as one of the quantities a, |3, y, would be known, not 

 only the other two but likewise a b would be known. For ob- 

 taining the knowledge of this one quantity the equations be- 

 tween time and place in the ellipse may be used. 



If we conceive a circle to be described about the centre of 

 the ellipse, whose radius is equal to the semiaxis major of the 

 ellipse, and if we designate as corresponding points in the 

 circle and the ellipse, those which have in both curves the 

 same abscisses and corresponding ordinates, all areas of the 

 two curves, inclosed by the radii of the corresponding points, 

 by the chords between them, and by the arcs of the curves, 

 will be in the ratio of the axes. The sector of the circle 



corresponding 



