The EUrptic Functions Defined. 



75 



Let U and T', (Fig. 6) be the terminations of two arcs of 

 the circle of the argument, of lengths respectively denoted by ^ii 

 and 2y, and let P and J be the corresponding points of the 

 circle of the amplitude. A circle of the interior system is 

 drawn, tangent to the chord AJ. Then if W and Z are the 

 terminations of arcs 2u + 2v, and 2m — 2r, they correspond 

 respectively to 31 and X, the extremities of chords drawn from 

 P, tangent to the same interior circle. I use, as before, r to 

 denote the radius of this circle, and — a for the abscissa of its 

 center, and designate the co-ordinates of P by c and s. The 

 line 3IX is then the same as that the equation of which was 

 written, in equation 6. We have seen that we can express the 

 functions sn {u + v) and oi {u -{- v) in terms of the abscissa 

 of the point M. In obtaining the abscissa from the equation, 

 we shall necessarily find at 

 the same time that of the 

 point JSf, and thus derive 

 simultaneously the formulae 

 for addition and subtraction 

 of two arguments. 



To eliminate y between 

 the equation of 3IjV and ^ 

 that of the circle of the am- 

 plitude we have to replace 

 that number by V-R- — x- ; 

 and if at the same time we 

 put V^^ — C" fors we have: 



{d'c + 2aR' + cR") x + (i?' — a") V {R' — 

 + R' {a- + 2ac + R' 



Fiff. 6 



- 2r-') = 0. 



On clearing of radicals, collecting terms, and dividing by R-, 

 this equation is readily reduced to the form, 



< a' + 2ac -\-R'y- x' + 2x ( a-c + 2aR' + cR' ) ( a- + 2ac + R' — 2r- ) 

 + ( a-c + 2aR' +cR~y~ 4r'i?- ( a" + 2ac + R' ) 

 + 4r*i?- = 0. 



