The EUlpiie Functions Defined. 



79 



In order to deduce the formula for dn {u + v), let us con- 

 sider another diagram, (Fig. 7), in which, in so far as the same 

 letters are used as in the 

 preceding, they stand for 

 the same things; viz., the 

 double arguments 2u and 

 2v terminate at U and V 

 respectively, and these 

 points correspond to P and 

 to J. Let us name the 

 arcs AP and AJ, 2^ and 

 2v'' respectively, and let an 

 arc AJ', of a length 2v'', be 

 measured from A in the 

 negative direction. Join 



PJ', and draw the circle of the interior system tangent to this 

 chord. As this circle is not the same as was drawn in Fig. 6, 

 its centre is marked c' instead of c. We also denote the abscissa 

 of c' by — a', and the radius of the new cirle by r'. In the 

 former figure a chord tangent to the interior circle would always 

 terminate in points such that their corresponding points on the 

 circle of the argument were separated by an arc of 2r; here 

 that constant arc is of length 2 {ii -{- v); so, while there we had 



for en V, and — — , — - for dn v, here we shall have 



R -{- a ' R + a 



- == en (u -\- v), and 



R — a' 



- = dn{u 4" v). 



(29 



R + a' V ' " R + a 



Let S be the middle point of the chord PJ', and T its point of 

 contact with the interior circle; draw CS and c'T, which will be 

 perpendicular to PJ', and CI parallel thereto. 



Now as J'P = 2^ + 2<', the angle SCJ' = s^ + v'S and if from 

 this be taken ACJ' or 2v'', the remainder, ACS, is <p — 4'- Now 

 cTisr', but 



