Manchester Memoirs, Vol. Ixiv (1920), No. 1 5 



r\ and r 2 are the focal distances of P„ P 2 , a is the semi-transverse axis, 

 and ix is the acceleration at unit distance. 



As before, let ^ T , ^ denote the parameters of JP It P v 

 Then r T + r 2 = ae cosh iU/I + a«; cosh ^ 2 - 2a 



= 2 ae cosh \ (^ + /x 2 ) cosh \ (^ - ^ 2 ) — 2a. 



Now let .P be the point whose parameter is ^(/xi + /x 2 ) and N the foot 

 of the corresponding ordinate. P will not now be the middle point 

 of P r P 2 . 



■ '• r x + r 2 = 2 e. CN cosh £ (^ - ^) -2a. 



Let ^V be taken on the axis of x so that CN' = a + r= a + e. CN - 

 a = e. CN. 



.'. r T + r 2 =2 CN ' cosh J (^ - tt 2 ) - 2a. 



If /*'„ jP' 2 are taken on opposite sides of P so that their parameters 

 fin fi's satisfy the relations fi x + fi 2 = 2^ , i fj! x — fi z = fix—fta, 



we easily find r T + r 2 = AN\ + AN\ , (n) 



N\,N\ being the feet of the ordinates from Z 3 '„ P ' 2 , 



and k = N\N\ (12) 



. . £ / - f , =% , 2 ^cosn fix— i) - smn £ ^ z 



sinh I </>!, 

 and similarly J /— — — — = sinh J u y 2 = sinh £ </> 2 . 



Hence p'„ /x' 2 are geometrical representations of <£„ <£ 2 . 



, /a*_ 

 J a? 



Now / j£ z =—fiT + fi 2 + e sinh ^ - <? sinh ^ 



= — ^ + </) 2 + sinh <£ z — sinh <£ 2J on reduction. 



That the analogy between the two cases is complete will be more 

 fully appreciated when it is recollected that in the case of the ellipse, 

 ju, besides denoting the angle ACQ, is also a measure of the sectorial 

 area ACP. In fact the area ACP'is equal to J abu, both in the ellipse 

 and hyperbola. In the case of the ellipse the parameter of P is chosen 

 half way between those of P x and P 2 and those of P\ P' 2 are so chosen 

 that jtA 1 — /x 2 = /xi — ju,2> ni + /i S = 2jn. 



Routh's " Dynamics of a Particle," p. 226. 



