4f4 Prof. Encke on the Calculatiofi 



of the distance of the focus from the centre, we have, denotuig 

 the semiaxis of the true ellipsis by a' and b', and the angle of 

 excentricity of the true ellipsis by !|!' : 



— - = sin <p' = — ^ — J •' 



(31) ^ « 



^ = cos a' = — r • 



I 



One ellipsis being a projection of the other, we have, calling 

 the inclination of the two ellipses to each other V : 



(S2) a'h' cos i' = ah. 



If we designate the angle between the line of nodes and the 

 perihelion counted in the true ellipsis by w', and the angle 

 formed by the line of intersection of the two planes with the 

 original principal axis in the plane of projection by 83 , we 

 have for the coordinates of the perihelion from the centre 

 these equations : 



,gg, a' cos w' = — / cos (P — ft) 



^ ' a sin 'J cos i' ■= —I sin (P— ft). 



Lastly, we have for the common radius of the line of nodes, 



(34) -jr: sin<y'-+ -^cosi)'^= -— sin (ft— 4,)^+ -^ cos (ft-(i>)', 

 o - a 0- a 



and for the two radii perpendicular to the line of nodes, the 

 one of which is the projection of the other : 



(35) ( jjr cos w''^ + 75-sin w'-) sec ?'- = -j^ cos (ft— w)® 



4 — 5- sin (ft— w)^ 

 a- 



In these five equations the conditions of the problem are con- 

 tained. 



Combining the squares of the equations (33) with (34) and 

 (55), and duly applying (32), we obtain by simple addition 



6'- sin w^ + h'^ cos w'^ cos ?'- = a^ + Ir — /-, 

 and this equation combined again with the sum of the squares 

 of (33) gives 



(36) b'- + b'^ cos i''^ = a^ + b^ - R^ 



This equation squared and combined with the equations (31) 

 and (32) will assume this form : 



b" sin i'* = (a* + b^--R^Y- ^a" ¥ + ^a^ b^ -^ 

 or substituting the value of / 



(37) 6'*sini'« = {a'— 6'— R-cos2(P-4,)} + R« sin 2(P-<ij)». 



The 



