821 the simultaneous values ?n. = 0, ?i = being now admitted in the numerator, although still excluded from the denominator, then if we write for instance ?t + 2o> instead of it, each factor in the numerator is changed into a contiguous factor, and the numerator remains unaltered, except that we introduce certain factors which lie outside the bounding curve, and omit certain factors which lie inside the bounding curve ; we in fact all ect the result by a singly infinite series of factors belonging to points adjacent to the bounding curve; and it appears on investigation that we thus introduce a constant factor exp y(u + v). The final result thus is that the product does not remain unaltered when u is changed into u + 2u, but that it becomes therefore affected, with a constant factor, exp y(u + ca). Arid similarly the function does not remain unaltered when u is changed into u + 2v, but it becomes affected with a factor, exp S(u + v). The bounding curve may however be taken such that the function is unaltered when u is changed into u + 2ca , this will be the case if the curve is a rectangle such that the length in the direction of the axis of vi is infinitely great in comparison of that in the direction of the axis of n ; or it may be taken such that the function is unaltered when u is changed into u + 2v, this will be so if the curve be a rectangle such that the length in the direction of the axis of n is indefinitely great in comparison with that in the direction of the axis of m ; but the two conditions cannot be satisfied simultaneously. _n. We have three other like functions, viz., writing for shortness m, n to denote ??i + i, n + J respectively, and (m, n) to denote mu + nv, then the four functions are " nn nn nn the bounding curve being in each case the same ; and, dividing the first three of these each by the last, we have (except as to constant factors) the three functions sn u, en u, dn u; writing in each of the four functions u+ 2w or u + 2u in place of u, the functions acquire each of them Hie same exponential factor exp y(u + u), or exp S(u + v), and the quotient of any two) of them, and therefore each of the functions sn u, en u, dn u, remains unaltered. It is easily seen that, disregarding constant factors, the four 0-functions are in fact one and the same function, with different arguments, or they may be written Ou, 6(u + ^ca), 6(u + ^v), 0(u + o> + Ju) ; by what precedes tho functions may be so determined that they shall remain unaltered when u is changed into u + 2ca (that is, be singly periodic), but that the change u into u + 2v shall affect them each with the same exponential factor exp 80 + i>). 12. Taking the last-mentioned property as a definition of the function 6, it appears that may be expressed as a sum of exponentials 0u = A2 cxp 2?(wn2+m) , where the summation extends to all positive and negative integer values of m, including zero. In fact, if we first write herein + 2w instead of u, then in each term the index of the exponential is altered by 2um, =*2miri, and the term itself thus remainsunaltered; that is, 6(u + 2u) = On. But writing u + 2v in place of u, each term is changed into the succeeding term, into the factor exp (u + v); in fact, making the change in question u into u + 2t>, and writing also m - 1 in place of m, vm?+ um becomes v(m - 1 ) 2 + (u + 2v)(m - 1 ), - vm? + um -u- u,andwethushave0(z{ + 2i/) = expf - (it + v)} .611. L (a J In order to the convergency of the series it is necessary that irivin? exp -- - should vanish for indefinitely large values of m, and iv this will be so if be a complex quantity of the form a + j3i, a nega tive ; for instance, this will be the case if u be real and positive and v be = i into a real and positive quantity. The original definition of d as a double product seems to put more clearly in evidence the real nature of this function, but the new definition has the advantage that it admits of extension to the 0-functions of two or more variables. The elliptic functions sn u, en u, dn u, have thus been expressed each of them as the quotient of two 0-functions, but the question arises to express conversely a 0-function by means of the elliptic functions ; theformis found to be 6u = C exp (Aw 2 + B/^ sn 2 u duty , viz., Qu is expressible as an exponential, the index of which depends on the double integral f ~f a sn 2 u du" 2 . The object has been to explainthe general nature of the elliptic functions sn u, en u. dn u, 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 application 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 func tion the ratio of the two periods 2o>, 2u is imaginary shows that we cannot have a function of a single variable, which shall bo triply periodic, or of any higher order of periodicity. For if the periods of a triply periodic function $(u) were 2w, 2v, 2^, then m, n, p being any positive or negative integer values, we should have <j>(u+ Zulu + 2>iv + 2px) = <t>u ! w v, x must be incommensurable, for if not, the three periods would really reduce themselves to two periods, or 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 m<a + nv+2 ) X shall be each of them as small as we please, say <i>(u + e) = <pu, and thence <f>(u + k() = (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 <f>u = a constant, or at least the function if not a constant is u 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 $(u, v) having for u, v the simultaneous periods 2a>, 2u ; 2n, 2i/ ; 2x, 2x ; 2iJ/, 24/ ; or what is the same thing, such that m, n, p, q being any integers whatever, we have <f>(u + 2mu + 2)iv + 2px + 2tjty, v + 2i>ua + 2nv +2px + 2qfy ) = (f>(u, r) ; and similarly a function of 2)t variables may be 2ft-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 dx <1x x< . + -7= = du. -,- V X v Y V yfy T A/^7^^ 1 we regard certain symmetrical functions of x, y, in fact the ratios of (2 4 =) 16 such symmetrical functions as functions of (u, v); say we thus have 15 hyperelliptic functions f(u, v} analogous to the 3 elliptic functions sn u, en u, dn u, and being quadruply periodic. And these are the quotients of 16 double 0-functions 6(u, r), the general form being 0(u, r) = A22 exp {tfa,h,b)(m,n)-+mu+nv}, where the summations extend to all positive and negative integer values of (m, n) ; and we thus see the form of the 0-function for any number of variables whatever. The epithet "hyperelliptic" is used in the case where the differentials are of the form just mentioned dx 7^., where X is a rational and integral function of x; the epithet " Abelian" extends to the more general case where the differential involves the irrational function of x, determined by any rational and integral equation <f>(x, y) = whatever. As regards the literature of the subject, it may be noticed that the various memoirs by Riemann, 1851-1866, are republished in the collected edition of his works, Leipsic, 1876 ; and shortly after his death we have the Theorie dcr Abclschcn Fundioncn, by Clebsch and Gordan, Leipsic, 1866. Preceding this we have by MM. Rriot and Bouquet, the Theoric dcs Fondions doullcmcnt p&nodiques ct en 2)(trticulicr dcs Fondions Elliptiqucs, Paris, 1859, the results of which are reproduced and developed in their larger work, Thcoric dcs Fondions Elliptiqucs, 2 ed., Paris, 1875. 14. It is proper to mention the gamma (r) or n function, T(7i + l) = n?i, =1.2.3. . . ?;, when is a positive integer. In the case just referred to, n a positive integer, this presents itself almost everywhere in analysis, for instance, the binomial coefficients, and the coefficients of the exponential series are expressible by means of such functions of a number n. The definition for any real positive value of n was taken to be Tn J^ * x n ~ l ^~ dx; it is then shown that, n being real and positive, r (n + 1) = nTn, and by assuming that this equation holds good for positive or negative real values of n, the definition is extended to real negative values ; the equation gives rl = OFO, that is TO = 00, and similaily r( - n) = oo , where - n is any negative integer. The definition by the definite integral has been or may be extended to imaginary values of n, but the theory is not an established one. A defini tion extending to all values of n is that of Gauss Un = j k", the ultimate value of k + k limit. -- -*- - n + l.n + 2.n + 3 . . . n being = ; but the function is chiefly considered for real values of the variable. A formula for tho calculation when x has a large real and positive value is TIx = V2ir x* + l exp ( - x + j^. + ...), or as this may also be written (neglecting the negative powers of .r) rir = V2^ exp {(.r + $) log x-x}.
