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



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

 and {a is the acceleration at unit distance. 



As before, let ^ ^ denote the parameters of P lt P 3 . 

 Then r x + r 2 = ae cosh ^ + ac cosh ^ 2 - 2a 



= 2 ae cosh J (^ x + /* 2 ) cosh J (^ - /jt? )— 2a. 



Now let /* be the point whose parameter is J^ + tta) and iVthe foot 

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

 of P X P V 



. ". r T + r 2 = 2 £. CiV COsh I (/xr - /x 2 ) - 2a. 



Let TV' be taken on the axis of x so that CN' = + /- = a + *. C7V- 

 • *• fi + ^2 = 2 CiV ' cosh \ (/xj - ^ 2 ) - 2a. 



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

 p' M M ' 2 satisfy the relations /x'i + // 2 = 2//, //i — /x' 2 = /x i — /x 2 , 



we easily find r I + r 2 = -4iV , I + ^4iV T, 2 , (11) 



JV'jyN ' 2 being the feet of the ordinates from Z 3 '„ /> ' 2 , 



and A = N\N , a (12) 



i / r I + r 2 + ^ _ 1 /2^/V ', / 2 / cosh ■ T \-ci nn l '- 



. 2 v — - *J—^- =ij Mcosh^-iJ-smhJ^- 



sinh ^ (/>!, 



/^ + r 2 - £ 



and similarly J / — — = sinh J n' 2 = sinh \ <f, 2 . 



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



Now / / — = — /xi + /x 2 + ^ sinh ^ - <r sinh iX *. 



= -</>i + (/)2 + sinh c^ — sinh <£ 2 , 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, 

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

 area ACP. In fact the area A CP is equal to \ 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 /a 1 — /a 2 = /xi - /X 2 > /A I + /A 2 = 2 jX ■ 



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



