426 Mr. A. Cayley on the Theory of Elliptic Motion. 



And writing 



d6' 



=r'=p, 



= rW' = q, 



we have 7-'=p, 6'= -^, and T= nKP^-'f- ^)) whence putting 

 H = T-U, the value of H is 



And by Sir W. R, Hamilton's theory, the equations of motion are 



Or substituting for H its value, the equations of motion are 

 'dr 



Tt=P' 



_ ? 



dt~ 1^ '^ r'^' 



Putting, as usual, ^ = n?a?, and introducing the excentric 

 anomaly u, which is given as a function of t by means of the 

 equation 



nt + c'=u — esiuM 



( so that -jT = 



- ), the integral equations are 



dt 1 — ecos 



q = na^ \/\ — e-, 

 nae sin u 

 1 — ecosM 

 /•=a(l— ecosw), 



^-i;7=tan-'(:^^lEZ£i^); 

 V cosM — e / 



where the constants of integration a, e, c, ■or denote as usual the 

 mean distance, the excentricity, the mean anomaly at epoch, and 

 the longitude of pericentre. 



