The Elliptic Functions Defined. Tl 



tions of u and v, for which the following values are furnished 

 by the preceding equations, (15) to (18), and (21), 



R — c o R + c 



' cn'u ~ 



2R 2R 



(R-\-a)- — r- r , R — a 



sn^v = — z-=- — ; 7, — en V = ^^^ — ; an v = 



also k- = 



(25 



{R -\- af R -\- a R -\- a 



4: aR 



{R+cif — i-' 



No value has hitherto been found for dji u, in our present nota- 

 tion, — i. e., where 2u corresponds to an arc of amplitude termi- 

 nating at c, s; — hence we must compute one from the formula 

 dnhi = 1 — k-sn-u. We have 



, ., ^ AaR R^c 



{R-\-a)- — r' 2R g 



_ 2a{R — c)_ a- + 2ac + i?' — r' ^ 



~ ~ {R + aY^r- ~ {R + a)' — r" 



The substitution of these values in the foregoing fractions 

 presents little difficulty. Beginning, in the first fraction, with 

 the term r-{AR — J5), we have to evaluate in the first place the 

 factor AR — B, by substituting for A and B the polynomials 

 which they replace. We thus find this factor equal to {R — c) 

 {R — ay. We recognize in this the product of the numerators 

 of the above values of sn-ii, and dn~v, and in the other factor, r", 

 that of cn-v. The denominators, all multiplied together, amount 

 to 2R {R -}- ay , and by this quantity, therefore, we will divide 



both the numerator and denominator of the value of — ^-^ — In 



the last term ot the numerator we now have ^ — ^^ „ / ' , — ^ 



AR yR -f- a) 



But AR -\- B directly reduces, (as above, but with change of 



A 7? -J— R 

 sign) to (-R -{- c)(R -\- ay. Hence „ , — rr, is cn~u, and there 



ZR{^R -\- a)~ 



i.1 £ i. ^ — ^'" «^ + 2ac -\-R- — ?'\ , , , . . 



remains the lactor , ^ , -r , or -:=-- r:? ; and this is 



{R -f- ay (R -j- ay 



evidently strv dnru. 



