Royal Astronomical Society, 389 



eafch determination, M. Villarceau arrives thus at the following 

 equations : — 



i . — — — =0, or S . i^^ = 

 D« D« 



eifF '"• >^^^ 



S.^=0, or 2.^-^=0 '' '' 



&c., 



and he shows how, supposing an ellipse roughly drawn by hand, 

 the value of D may be found graphically ; and it will then be pos- 

 sible to solve the equations. 



The projected ellipse being thus determined, the real ellipse will 

 be found from the consideration that the origin of co-ordinates is 

 the projection of the focus of the real ellipse, while the centre of 

 the observed ellipse is the projection of the centre of the real ellipse. 

 The formation of the corresponding equations is a not difficult pro- 

 blem of analytical geometry. This transformation, however, is not 

 required till all the other o))erations are completed. 



The points determined by observation are not generally found 

 exactly upon the projected ellipse. In order to have points upon 

 the ellipse which shall be the subjects of further investigation, 

 M. Villarceau transfers the observed points to the ellipse by draw- 

 ing normals to the ellipse, and taking, instead of the point actually 

 determined by observation, the foot of its normal. If x' and y' be 

 the co-ordinates determined from observation, x and y those of the 

 foot of the normal, then 



dF 

 , dx 



\dx) \dy ) 



F(^'.y') 



dF 

 dy 



y=y'- '^ . F (x', v') 



\dx) \dy) 



with sufficient exactness. 



The next point is, to introduce the consideration of time ; and 

 this is to be done by making the areas described by the radius 

 vector in the projected ellipse proportional to the time. The areas 

 can be expressed in terms of the corrected co-ordinates and the 

 constants without much difficulty, the whole of these admitting of 

 further correction if necessary. M. Villarceau remarks that if there 

 are four observed places, the solution of the four equations F {x, y) 

 = will give four of the quantities a, b, c, d, e, in terms of the 

 fifth ; that these four observations will give three areas between 

 which there are two equations of proportion ; and that thus, besides 



