TRANSACTIONS OF THE SECTIONS. 17 



The coordinates of any point in the hyperbola referred to its axes may be repre- 

 sented by 



a?=«csh (u), 



y=as/ e i — lsnh (m). 



If u, u' denote the values of u corresponding to the two extremities of the arc, 

 we have 



r=«(e csh (V)— 1), r'=a(e csh (w 1 ) — 1), 



or 



2« 



/i 2 =a 2 [csh («)— csh ( M ')]^+a'-( e 2 _l) [snh («)— snh (w')] 2 



Also twice the area of the sector limited by r and r' 



=« 2 Ve 2 — l[(esnh u — ii) — (esnh?«'— u')~] 



=fl2 V^Ti[2( e csh ? i+^') sn h^-( M - M ')~l, 



and twice the area described in a unit of time is 



yV«(e a — 1). 

 Hence 



t= (fL 8 ) 4 [ 2 ( e cd£+*) snh^ - (« - «')] ; 



and therefore if a be given, then r+r', k, and t are functions of the two quantities 

 ecsh w "t~ M and u—u'. 



Let M-M'=2a,and e csh n j" M ' = csh (/3), which is always possible since c is 



greater than 1. 

 Then 



!±^=csh(j3)csli(*)-l, 

 la 



—= snh (/3) snh («) ; 



therefore r +>''+ k - cs h (0 + a) - 1 

 2a 



and 



!^Z_=*=c8h(/3-«)-l 



2a 



Also 



;= fc!\*[2 csh (0) snh (a) -2a], 

 = (f-\* [snh +a )-(/3-r-a)- snh (/S-aH/S-a]. 



As before, the first two of these equations give /?+ a. and — as in terms of r+r'+tc 

 and r+r'— ifc, and the last equation is the expression of Lambert's theorem in the 

 case of the hyberbola. 



