Prof. Sylvester's Astronomical Prolusions, 53 



one a sort of verification by aid of trigonometrical formulae in 

 which the eccentric anomalies are introduced; a second of a 

 similar nature, but dealing only with the true anomalies; a 

 third founded on a property of integrals*; and a fourth, 

 perhaps the most remarkable of any, derived from the general 

 expressions for the time in an orbit described about two centres 

 of force varying according to the law of nature, but one of 

 them supposed to be situated in the orbit itself, and to become 

 zero. Notwithstanding this plethora of demonstrations I ven- 

 ture to add a seventh, the simplest, briefest, and most natural 

 of all, in which I employ a direct method to prove, from the 

 ordinary formulae for the time of a planet passing from one 

 point to another, that, when the period is given, the time is a 

 function only of the sum of the distances of these points from 

 the centre of force, and of their distance from one another. 



Let p, p 1 be the distances of the two positions from the sun, 

 c their distance from one another, v, v' the true, u, u 1 the eccen- 

 tric, m, m l the mean anomalies thereunto corresponding, e the 

 eccentricity, (o = m—m , J s=p + p , } A = |(s 2 — c *)> * nen 

 /o=l— ecosu, /o'=l— ecosw', mz=u—esmu, m'=w'~esinw', 



pcosv = cosu— e, p sin v = \/ 1 — e 2 sin u, 

 p f cos v' = cos u' — e, p' sin tf = \/l— e 2 sintt'> 

 c*=p*+p'*-2pp'cos{v ! -v). 



Writing for brevity p, p\ q, q' for cos u, cos v!, sin u, sin u\ 

 we have 



s=2—ep,—ep t , co=u—u'—eq-\-eq f , 

 A=pp' + pp f cos (v'—v) = 1 +/?, p 1 + q,q' — 2e(q + q 1 ) + 

 e*(\-q,q'>-pp'). 



Let J= ., ' ' ,; ; then J is the determinant, 

 d\e } u, u) 



t+2e(i+ i y-< Z9 ')J' l-* 2 W+A)J' \-e*(pq'+p 1 q) J 



—p-p' ; eq ; eq' 



— q + q' ; l — ep ; — l+ep'. 



* The property in question, discovered by Lagrange, is that the integral 



rdr 



£ 



_. VH+Mr+Nr 2 

 may be transformed into 



(y 2 +h)dy 



J {x*+h)dx _ f 



V a+ &»+ ex 2 + dx d 4- ex 3 J V«- 



j+6y+ey 2 4-c?y 3 +ey 4 ' 

 in applying it to Lambert's theorem a, 6, c are made to vanish. This 

 transformation and its consequences appear to us to deserve further study ; 

 as far as I know it has not been touched upon by the writers on elliptic 

 functions. 



