The Elliptic Functions Defined. 71 



Another geometrical definition of dn ii may be obtained as 

 follows. Let J/ be any point on the circle of the amplitude, the 

 termination of an arc 2cr', and let its coordinates be x' and y'. 

 The diameter FA is divided at L, the foot of the ordinate, into 

 the segments R + .r' and R — x', which are proportional to the 

 squares of the chords FM and AM, and hence to the squares of 

 cos cr and sin e respectively; hence (if the arc 2ii' correspond to 

 2t.'') we have 



or u' = — 975^ ^^^ s;r ii = — no— (21 



and from the latter equation 



x' = R {I — 2sn-u'). 



Now the square of the distance of any point x', y' from the 

 point c is 



y" + x" + 2rf.r' + (r 



whence, if /' denote the length of the tangent to the circle whose 

 centre is c, 



r' = 7/'- + x" + 2rtJ-' + o' — r- . 



But the point being M, on the circle of the amplitude, we may 

 put for y'^ + '-x'' the equivalent R~, and for x' in the succeeding 

 term the value just obtained, and find 



r^ = K- + 2aR (1 — 2s»-m') + cr — r 

 r- = i?^ + 2((R + (r — r- — AaRsjfn' 



or finally 



r = [ , iJ + aY - ,■'.] (1 - ^^j^^f— . s„= w) . 



The former factor is the square of the tangent AT, as given in 

 a former equation, (14), and the latter is, — see equations (18) 

 and (19),— 



1 —- Jcsif ii' or chru. 



Hence the value of f is briefly expressed 



t' = idnu'. (22 



