The Elliptic Functions Defined. 81 



may be connected with the methods of the calculus, and also 

 the character of the assumption which has to be made, in order 

 to give to the circle of the argument its appropriate relative 

 magnitude, i. e., to fix the value of K. 



By subtracting, one from the other, the two formulae in- 

 cluded in equation (27) we have 



, 2sn r en u dn u 



sn (u -r- v) — sn iu — r ) = -z, -t, — 5 5— • 



^ ^ 1 — k- snu sn'v 



Let us replace u by ic + \h and v by \li, and then divide each 

 side of the equation by h ; and we obtain 



sn {lo -\-h) — sn ic _ 2sn (^h) en (tv + ^h) dn (iv + ^h) 

 h ~ [ 1 — fc' sn' {IV + |/i) sn' (^/i) ] h 



_ en (w -\- ^h) dn (iv + ^h) sn (^h) 

 "~" 1 — A;"-S7i"(ty + ^/i) S7i'-(^/i) ' ^/i 



We have here separated the result into two factors. When 

 h is made to approach zero, the denominator of the first factor 

 approaches unity, because the second term of that denominator 

 contains the vanishing factor snH^h). The numerator of the same 

 factor approaches en iv dn u\ But in the second factor we have 

 the ratio of the chord of a small arc of the circle of the ampli- 

 tude divided by the corresponding arc in the circle of the argu- 

 ment. That this factor may approach unity as h approaches 

 zero, it is necessary that corresponding arcs, measured from the 

 origins in the two circles, shall approach equality as their length 

 is diminished. Or if corresponding arcs are traced simultane- 

 ously by two points, one on each circle, these points mufet set 

 out from the origins with equal initial velocities. This condi- 

 tion will determine the relative magnitude of the circles; and 



1 -. • ^n^■, ■, ^ i- -, p , /^ P SH ( IV 4- Jl) Sn IV 



when it IS tulfilled, the limit, tor h = 0, or — ^ 



is en IV dn iv. 



