70 Colorado College Studies. 



the ratio in which the chord of 2(f is divided at the point of con- 

 tact of its tangent circle of the interior system; or as the ratio 

 in which the diameter of the circle of the amplitude is divided 

 at the center of the same interior circle. That is, 



TP cF R — a ^^_ 



We may also express k" in terms of the same letters, — not 

 forgetting, however, that k' is constant, while a and r vary with 

 u. The length of CD as found from equation (11), is 



R' + cr — r- 



CD = 



2o 



For AD, we must add R, and then divide 2i? by the result to 

 obtain /r, whence 



^' - iR + ay^=^^- ^^^ 



But from the above value of dn u, it appears that 



2R 2a 4:aR 



(1 + (hi u) (1 — dn u) 



R -{- a R+a {R+aY 



whence 



1 — dirn _ 4:aR 



sifu (R -\- (lY — ?•-' 



Comparing this with the foregoing value of /r we obtain the 

 equation 



1 — dii-u _ ,2 



s — /v" 



sn-u 

 or 



dnhi = l — k'sifu. (19 



The definitions of the three "ellij^fic finicfions,'' as stated in the 

 analytic form, are accordingly as follows: 



SU U = sin cr; Ol It = COS cr; du U = Vl — k' silT cr . (20 



The first and second of these definitions determine the sign as 

 well as the magnitude of sn it and cit u: the sign of dit u is 

 taken always positive. 



