262 



Mr C. Taylor, On the geometrical 



[May 19, 



to be unnecessary, for the theorem may be proved by a purely 

 geometrical process which presents no serious difficulty. 



We have to prove that : 



The time in an arc of an elliptic orhit described about a focus 

 may be expressed in terms of the major axis, the chord of the arc, 

 and the sum of the distances from that focus to the extremities of the 



arc. 



I. 



Let S be a focus of an ellipse, and AA' and BB' its axes. 



Take any fixed diameter PGP' and let CD be the conjugate 

 semi-diameter. Let P8 meet CD in K and the tangent at P' in 

 H. Then PK is equal to CA and P^to AA\ 



Also, if CN be the abscissa of P, 



,S^ : CK= C8 : CA, 

 or briefly SK=e.CN, 



if e be used to denote the eccentricity. 



With 8 as focus and P and H as vertices describe a second 

 ellipse, and let KL be its semi-axis conjugate ; then, if 8' be the 

 further focus of the first ellipse, 



CD^=8P.P8' = 8P.8H=KL\ 



In the second ellipse place any principal double ordinate gq' bi- 

 sected in ; and in the first ellipse let QOQ' be that double ordinate 

 of PP' which passes through a. Then we have to shew : 



(i) Th&t QQ' = qq. 



(ii) Thsit8Q + 8Q' = Sq + 8q. 



