787] 
FUNCTION. 
533 
The elliptic functions sn u, cn u, dn u, have thus been expressed each of them as 
the quotient of two ^-functions, but the question arises to express conversely a 0-function 
by means of the elliptic functions; the form is found to be 
viz. On is expressible as an exponential, the index of which depends on the double integral 
The object has been to explain the general nature of the elliptic functions smq cnu, dmi, 
and of the 0-functions with which they are thus intimately connected; it would be out 
of place to go into the theories of the multiplication, division, and transformation of 
the elliptic functions, or into the theory of the elliptic integrals, and of the applic 
ation of the 0-functions to the representation of the elliptic integrals of the second 
and third kinds. 
13. The reasoning which shows that for a doubly periodic function the ratio of 
the two periods 2<w, 2v is imaginary shows that we cannot have a function of a single 
variable, which shall be triply periodic, or of any higher order of periodicity. For if 
the periods of a triply periodic function 0 (u) were 2to, 2v, 2%, then m, n, p being any 
positive or negative integer values, we should have 0 (u + 2mw + 2nv + 2p%) = <f>u ; a>, v, % 
must be incommensurable, for if not, the three periods would really reduce themselves 
to two periods, or to a single period ; and being incommensurable, it would be possible 
to determine the integers m, n, p in such manner that the real part and also the 
coefficient of i of the expression mw + nv + p% shall be each of them as small as 
we please, say 0 (u + e) = <j>u, and thence 0 (u + he) = <f>u (k an integer), and ke as 
near as we please to any given real or imaginary value whatever. We have thus the 
nugatory result <fiu = a constant, or at least the function if not a constant is a function 
of an infinitely and perpetually discontinuous kind, a conception of which can hardly 
be formed. But a function of two variables may be triply or quadruply periodic— 
viz. we may have a function 0(it, v) having for u, v the simultaneous periods 2co, 2<w'; 
2v, 2v ; 2x, 2y' 5 2-0, 2\Jr'; or, what is the same thing, it may be such that, m, n, p, q 
being any integers whatever, we have 
0 (u + 2mw + 2nv + 2px + 2qy}r, v + 2mm + 2nv + 2p%' + 2qyjr') = 0 (u, v); 
and similarly a function of 2n variables may be 2?i-tuply periodic. 
It is, in fact, in this manner that we pass from the elliptic functions and the 
single 0-functions to the hyperelliptic or Abelian functions and the multiple 0-functions; 
the case next succeeding the elliptic functions is when we have X, Y the same rational 
and integral sextic functions of x, y respectively, and then writing 
+ ZXg = dv,