FUNCTION 823 satisfying the first of thcso conditions, then- 7 dx + -r- dy is a ay ax complete differential, audy = / ( - -,- dx + -7-7 dy ) 18. We have in what just precedes the ordinary behaviour of a function <p(x+iy) in the neighbourhood of the value x + iy of the argument or point x + iy; or say the behaviour in regard to a point x + iy such that the function is in the neighbourhood of this point a continuous function of x+iy (or that the point is not a point of discontinuity) : the correlative definition of continuity will be that the function $(x + iy), assumed to have at the given point K + iy a single finite value, is continuous in the neighbourhood ot this point, when the point x + iy describing continuously a straight infinitesimal element /t + ik, the point <f>(x + iy) describes continuously a straight infinitesimal element ( + i[i)(h + ik) ; or what is really the same thing, when the function (x + iy) has at the point x + iy a differential coefficient. 19. It would doubtless be possible to give for the continuity of a function <f>(x + iy) a less stringent definition not implying the existence of a differential coefficient ; but we have this theory only in regard to the functions <(>x of a real variable in memoirs by Riemann, Hankel, Dubois Reymond, Schwarz, Gilbert, Klein, and Darboux. The last-mentioned geometer, in his "Memoire sur les fauctions discontinues," Jour, dc V&ole Normalc, t. iv. (1875) pp. 57-112, gives (after Bonnet) the following definition of a con tinuous function (observe that wo are now dealing with real quantities only) : the function f(x) is continuous for the value x = x when h and e being positive quantities as small as we please, and any positive quantity at pleasure between and 1, we have for all the values of f(x 6h) -f(x) less in absolute magnitude than e ; and moreover J[x) is continuous through the interval x , x 1 (x,>x , that is, nearer +00) when /(.*) is continuous for every value of x between z and x lf and, h tending to zero through positive values, f(x Q + h) and (a; -A) tend to the limits /(a i ),/(x,) respectively. It is possible, consistently with this definition, to form continuous functions not having in any proper sense a differential coefficient, and having other anomalous properties ; thus if !, a. 2 , , . . . be an infinite series of real positive or negative quantities, such that the series 2 n is absolutely con vergent (i.e., the sum 2 a n , each term being made positive, is a convergent), then the function 2 n (sin nirx) is a continuous func tion actually calculable for any assumed value of x; but it is shown in the memoir that, taking x = any commensurable value ^ whatever, and then writing x= + h, h indefinitely small, the increment of the function is of the form (k + *)h , k finite, c an indefinitely small quantity vanishing with h ; there is thus no term varying with h, nor consequently any differential co-efficient. See also Kiemann s Memoir Ucbcr die Darstellbarkcil, &c. (No. xii. in the collected works) already referred to. 20. It was necessary to allude to the foregoing theory of (as they may be termed) infinitely discontinuous functions ; but the ordinary and most important functions of analysis are those which aro continuous, except for a finite number (or it may be an infinite number) of points of discontinuity. It is to be observed that a point at which the function becomes infinite is ipso facto a point of discontinuity ; a value of the variable for which the function becomes infinite is, as already mentioned, said to be an "infinity" (or a "pole") of the function ; thus, in the case of a rational function expressed as a fraction in its least terms, if the denomina tor contains a factor (x-a) n , a a real or imaginary value, TO a positive integer, then a is said to be an infinity of the m-th order (and in the particular case m = l, it is said to be a simple infinity). The circular functions tan x, sec x are instances of a function having an infinite number of simple infinities. A rational function is a one-valued function, and in regard to a rational function the infinities are the only points of discontinuity ; but a one-valued function may have points of discontinuity of a character quite distinct from an infinity : for instance, in the exponential function exp ( J where a is real or imaginary, the value ?t(=o; + ty)=a, is a point of discontinuity but not an infinity ; taking u = a + pc a% , where p is an indefinitely small real positive quantity, the value of the function is exp ( - e ~ ai J, = exp *- (cos o - i sin o), which is indefinitely large or indefinitely small according as cos a is positive or negative, and in the separating case cos a = 0, and therefore sin o = 1, it is = cos - i sin which is indeterminate. If instead of exp we consider a linear function A + Bcxp f -^ u - a C + D exp _ - 11 -a then writing as before u = a + pc ai , the value is = A ~ C, or = B -~- D, according as cos o is negative or positive. As regards the theory of one-valued functions in general, the memoir by Weieistrass, " Zur Theorio der eiudeutigen aualytischen Functionen," Bcrl. Abh. 1876, pp. 11-60, may be referred to. 21. A one-valued function ex vi termini cannot have a point of discontinuity of the kind next referred to ; if the representative point P, moving in any manner whatever, returns to its original position, the corresponding point F cannot but return to its original position, but consider a many-valued function, say an n- valued function a/ + iy , of x + iy ; to each position of P there correspond n positions, in general all of them different, of P . But the point P may be such that (to take the most simple case) two of the corresponding points P coincide with each other, say such a position of P is at V, then (using for greater distinctness a different letter "VV instead of V) corresponding thereto we have two coincident points (W), and ?i-2 other points W; V is then a branch-point (Verzweigungspnnkt). Taking for P any point which is not a branch-point, then in the neighbourhood of this value each of the n functions a/ + ?y is a continuous function of x + iy, and by what precedes, if P describing an infinitely small closed curve (or oval) return to its original position, then each of the corresponding points P describing a corresponding indefinitely small oval will return to its original position. But imagine the oval described by P to be gradually enlarged, so that it comes to pass through a branch-point V; the ovals described by two of the corresponding points P will gradually approach each other, and will come to unite at the point (W ), each oval then sharpening itself out so that the two form together a figure of eight. And if we imagine the oval described by P to be still further enlarged so as to include within it the point V, then the figure of eight, losing the crossing, will be at first an hour-glass form, or twice-indented oval, and ultimately in form an ordinary oval, but having the character of a twofold oval; i.e., to the oval described by P (and which surrounds the branch-point V) there will correspond this twofold oval, and 7i-2 onefold ovals, in such wise that to a given position of P on its oval there correspond two points, say P t , P 2 , on the twofold oval, and 7i-2 points P 3 , . . . P n , each on its own onefold oval. And then as P describing its oval returns to its original position, the point P/ describing a portion only of the twofold oval, will pass to the original position of P 2 , while the point P./ describing the, remaining portion of the twofold oval will pass to the ori-inal position "of 1 ; the other points P 3 , . . P./, describing each of them its own onefold oval, will return each of them to its original position. And it is easy to understand how, when the oval described by P surrounds two or more of the branch-points V, the corresponding curves for P may be a system of manifold ovals, such that the sum of the manifoldness is always = 7*, and to con ceive in a general way the behaviour of the corresponding points P and P . Writing for a momenta; + iy = u t x + iy v, the branch-points are the points of contact of parallel tangents to the curve tf>(u, v) => 0, a line through a cusp (but not a lino through a node), being reckoned as a tangent ; that is, if this be a curve of the order n and class in, with 5 nodes and K cusps, the number of branch points is = m + K, that is, it is 7i 2 - n - 28 - 2, or if p, (n-1) (n-2)-5-n, be the deficiency, then the number is = 2 - 2 + 2p. To illustrate the theory of the 7i-valued algebraical function x + i]/ of the complex variable x + iy, Eiemann introduces the notion of a surface composed of n coincident planes or sheets, such that the transition from one sheet to another is made at the branch points, and that the n sheets form together a multiply-connected surface, which can be by cross-cuts (Querschnitte), converted into a simply-connected surface ; the w-valued function x + iy becomes thus a one-valued function of x + iy, considered as belonging to a point on some determinate sheet of the surface : and upon such considerations he founds the whole theory of the functions which arise from the integration of the differential expressions depending on the Ti-valucd algebraical function (that is, any irrational algebraical function whatever) of the independent variable, establishing as part of the investigation the theoiy of the multiple 0-functions. But it would be difficult to give a further account of these investi gations. The Calculus of Functions. 22. The so-called Calculus of Functions, as considered chiefly by Herschel and Babbage and De Morgan, is not so much a theory of functions as a theory of the solution of functional equations ; or, as perhaps should rather be said, the solution of functional equations by means of known functions, or symbols, the epithet known being here used in reference to the actual state of analysis. Thus for a functional equation <px + Qy=$(xy taking the logarithm as a known function the solution is ^>x-c log x; or if the logarithm is not taken to be a known function, then a solution may be obtained
