530 
FUNCTION. 
[787 
where the accent denotes differentiation in regard to u ; and the addition-formulae : 
sn (u + v) = sn u cn v dn v + sn v cn u dn u, (4-), 
cn (u + v) — en ïccn v — sn u dn u sn v dn v, (4-), 
dn (u + v) = dn u dn v — k 2 sn u en u sn v cn v, (4-), 
each of the expressions on the right-hand side being the numerator of a fraction of 
which 
Denom. = 1 — k? sn 2 u sn 2 v. 
It may be remarked that any one of the fractional expressions, differentiated in regard 
to u and to v respectively, gives the same result ; such expression is therefore a 
function of u + v, and the addition-formulae can be thus directly verified. 
9. The existence of a denominator in the addition-formulae suggests that sn, cn, dn 
are not, like the sine and cosine, functions having zeros only, without infinities; they 
are in fact functions, having each its own zeros, but having a common set of infinities; 
moreover, the zeros and the infinities are simple zeros and infinities respectively. And 
this further suggests, what in fact is the case, that the three functions are quotients 
having each its own numerator but a common denominator, say they are the quotients 
of four ^-functions, each of them having zeros only (and these simple zeros) but no 
infinities. 
The functions sn, cn, dn, but not the 6-functions, are moreover doubly periodic; 
that is, there exist values 2co, 2v, = 4>K and 4 {K + iK') in the ordinary notation, such 
that the sn, cn, or dn of u + 2eo, and the sn, cn, and dn of il -f 2v are equal to the 
sn, cn, and dn respectively of u ; or say that </> (u + 2&>) = </> (u + 2v) = $11, where <£ is 
any one of the three functions. 
As regards this double periodicity, it is to be observed that the equations 
<f> (it + 2c0) = (f>u, (f> (u + 2a) = <f)U, imply (f)(u + 2moo + 2nv) — <f>u, and hence it easily follows 
that if &), v were commensurable, say if they were multiples of some quantity a, we 
should have cf>(u + 2a) = <f>u, an equation which would replace the original two equations 
$ (u + 2w) = <f>iL, (j) (u + 2v) = fat, or there would in this case be only the single period 
a ; to and v must therefore be incommensurable. And this being so, they cannot have 
a real ratio, for if they had, the integer values to, n could be taken such as to make 
2to« + 2nv = k times a given real or imaginary value, k as small as we please ; the 
ratio <w : v must be therefore imaginary, as is in fact the case when the values are 
4A and 4 (K + iK'). 
10. The function snw has the zero 0 and the zeros mœ + nv, m and n any 
positive or negative integers whatever ; and this suggests that the numerator of sn u is 
equal to a doubly infinite product, (Cayley, “ On the Inverse Elliptic Functions,” Gavib. 
Math. Jour. t. iv., 1845, [24]; and “Mémoire sur les fonctions doublement périodiques,” 
Liouville, t. x., 1845, [25]). The numerator is equal to 
uim (1 + —), 
\ TOft) + nv)