as before there is a least value for P. actually occurring in one or more periods, say in the period o' too-Hwa'; now take, if wet-w be a period, =N'r:#%. where N is an integer, and o''<: thenceus+vo'=uw-HN'(q'-woo)+"'w'; take then u-N'" NM-HA', where N is an integer, and As, is as above, and of:A'<\s; we thus have a period Na+N'a'-i-A'o-Hw'o', and hence, a period X'w-Hw'w', wherein A'<\e, "'<ro; hence "=o, and A =o.. All eriods of the form *-i-w' are thus expressible in the form NQ+N'a', where 0, Q are periods and N, N' are integers. But in fact any complex quantity, P+12, and in particular any other sible period of the function, is expressible, with u, v real, in the £ aw-Hwa'; for if w=p-Hia, o' = p +io', this requires '. P = up +"p’, Q = u2+vo', equations which, since w/o is not real, always give finite values for u and v. - It thus appears that if a single valued monogenic function of 2 be periodic, either all its periods are real multiples of one of them, and then all are of the form Mø, where 0 is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form. Na+N'o', where Q, Q are periods, and N, N'are integers. In the former case, utting r=2xiz/0, and the function f(z) = &G), the function oG) as, like exp (;), the period 2xi, and if we take t =exp (+) or {=A(t) the function is a single valued function of t. If then in particular f(s) is an integral function, regarded as a function of t, it has singularities only for t=0 and t=%, and may be expanded in the form 2.3.". Taking the case when the single valued monogenic function has :wo periods o, ø whose ratio is not real, we can form a network of parallelograms covering the plane of s whose angular points are the points c-i-ma-i-m'a', wherein c is some constant and m. m' are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of £ the function is at all capable occur for points of this primary parallelogram, any point, z', of the plane being, as it is called, congruent to a definite point, z, of the primary parallelom, 2’-z being of the form mos-Hm'w', where m, m' are integers. uch a function cannot be an integral function, since then, if, in the primary parallelogram |f(z)|<M, it would also be the case, on a circle of centre the origin and radius R, that |f(z)|<M, and therefore, if Xa.z" be the expansion of the function, which is valid for an integral function for all finite values of z, we should have |a. 4 MR", which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z. We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. Supposing the constant c used in naming the corners of the £ lelograms so chosen that no pole falls on the perimeter of a parallelogram, it is clear that the integral # f(z)dz round the perimeter of the primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter points as z, z+w (or as s, z+w') are mutually destructive. This integral is, however, equal to the sum of the residues of f(z) at the poles interior to the rallelogram. Which sum is therefore zero. £ cannot thereore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (s-a)*-i-(power series in z-a). that an ordinary point of f(z) is an ordinary point of 4,0s), that a zero of order m ##! in the neighbourhood of which £ has a form, (2-a)" multiplied by a power series, is a pole of 3(3) of residue m, and that a pole of f(z) of order n is a pole of 4×2) of residue -n; manifestly 4 (2) has the two periods of f(z). We thus infer, since the sum of the residues of p(z) is zero, that for the function f(z), the sum of the orders of its £ at points belonging to one parallelom, Zm, is equal to the sum of the orders of its poles, 2n; which is £y expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function f(z)-A, where A is an arbitrary constant, we have the result, that the function f(z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is #: every conceivable complex value does arise as a value for the doubly periodic function £ in any one of its parallelograms, and in fact at least twice. The number of times it arises is called the order of the function; the result suggests a property of rational functions. if A, denote the generalized logarithm, -w!A[f(zo-i-w')]—Af(se)]], that is, since f(zo-Ho’) = f(zo), gives 2xiNa, where N is an integer, similarly the result of the integration along the other two opposite sides is of the form 2 riN'w', where N' is an integer. The integral, however, is equal to 2ritimes the sum of the residues of z (# at the poles interior to the parallelogram. For a zero, of order m, of f(z) at z = a, the contribution to this sum is 2rima, for a pole of order n at z =b the contribution is -2 rinb; we thus infer that Ema-Znb-Na-HN'w': this we express in words by saying that the sum of the values of a where f(z)=Q within any parallelogram is equal to the sum of the values of 2 where f(z) = x save for integral multiples of the periods. By considering similarly the function f(z)-A where A is an arbitrary constant, we prove that each of these sums is equal to the sum of the values of 2 where the function takes the value A in the parallelogram. We pass now to the construction of a function having two arbitrary periods w, o' of unreal ratio, which has a single pole of the second order in any one of its parallelograms. For this consider first the network of parallelograms whose corners are the points a-ma-i-m'w', where m, m take all positive and negative integer values, putting a small circle about each corner of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by *, ***, *****'. Pot-w'; we shall denote zo, so-Hø by A. B., this parallelogram II* is surrounded by eight other parallelograms, forming with IIo a larger parallelogram III, of £ side, for instance, contains the points zo-w-to', zo-o', zo-w'+w, ze-w' +2e. which we shall denote by AI, B, C, D). This parallelogram II, is surrounded by sixteen of the original parallelograms, forming with II, a still larger parallelogram II, of which one side, for instance, contains the ints so-2a-2a.', zo-w-24', zo-2w', zo-Ho-2 w", ze-H2~-22', zo-H.32-2", which we shall denote by A., B, C, D. E2, Fs. And so on. Now consider the sum of the inverse cubes of the distances of the point P. from the corners of all the original parallelograms. The sum will contain the terms of which the general term is ultimately, when n is large, in a ratio of equality with 2g"n”, so that the series So is convergent, as we know the sum zn" to be, this assumes that p +o; if P be on Aobo the proof for the convergence of So-1/PA: is the same. Taking the three other sums analogous to S. we thus reach the result that the series o(z) = -22(2-0)-2, where n is mo-'-m'w', and m, m” are to take all positive and negative integer values, and z is any point outside small circles descri with the points 0 as centres, is absolutely convergent. Its sum is therefore independent of the order of its terms. By the nature of the proof. which holds for all positions of 2 outside the small circles spoken of. the series is also clearly uniformly convergent outside these circles. Each term of the series being a monogenic function of s, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the riods." ~'; for *(s-Hw) is the same sum as e(s) with the terms in a slightly different order. Thus p(z+*) = p(z) and e(z+w') = e(s). Consider now the function and periods, we obtain, since the sum of the residues A is zero, a doubly periodic function ''', that is, a constant; this gives the expression of F(z) refe to... The indefinite integral frtz)dz can then be expressed in terms of z, functions $(2-a) and their differential coefficients, functions f(z-a) and functions log a (2-a). § 15. Potential Functions. Conformal Representation in General-Consider a circle of radius a lying within the region of existence of a single valued monogenic function, u+iv, of the complex variable z, *.x+ iy, the origin z=o being the centre of this circle. If z=r E(i) = r(cost-Hi sing) be an internal point of this circle we have and the equivalence of the integrals holds for every arc of integration. Conversely, let U be any continuous real function on the circumference, Ug being the value of it at a point P. of the circumference, and describe a small circle with centre at P. cutting the given circle in A and B, so that for all £ P of the arc £ we have|U-Uel <e, where £is a given small real quantity. Describe a further circle, centre P, within the former, cutting the given circle in A' ##. and let Ö be restricted to lie in the small space bounded by the are A'P.B' and this second circle, then for all positions of P upon the # arc AB of the original circle QP is greater than a definite hnite £auty which is not zero, say QP"> #: Consider now the integra AP2B and the greater arc AB. It is easy to verify that, for the whole circumference, *a*-i-aux-Hby-Haz(x*-y")+2b,xy+ . . . . ... I - 1 (U cosé t sing *-#Jua. di £: * *-###". - 1 cos 267 f sin 26 2-1/###", "-####". In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation 0°W/öx*+3*/öy';... the series is thus the real part of wer series in 2, and is capable of differentiation and integration within its region of convergence. Conversely we may suppose a function, P. defined for the interior of a finite region R of the plane of the real variables x, y, capable of expression about any interior point x, y, of this region by a power series in x-xo, y-yo, with real coefficients, these various series being obtainable from one of them by continuation. For any region Ro interior to the region specified, the radii of convergence of these wer series will then have a lower limit greater than zero, and ence a finite number of these power series suffice to specify the function for all points interior to Ro. Each of these series, and therefore the function, will be differentiable; suppose that at all points of Ro the function satisfies the equation where o: 6P1 #+5. O, we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function by means of the equation (x, y) /öp,..., op.,,, :- Q-f"(#-#"). The functions P, Q, being given by a finite number of power series, will be single valued in Ro, and P-HiQ will be a monogenic function of z within Ro. In drawing this inference it is £ that the region Ro is such that £ closed path drawn in it is capable of being deformed continuously to a point lying within Ro, that is, is simply connected. Suppose in particular, c being any point interior to Ro, that P approaches continuously, as z approaches to the boundary of R, to the value log r, where r is the distance of c to the points of the perimeter of R. Then the function of z expressed by * = (z-c) exp (-P-iQ) will be developable by a power series in (2-zo) about every point zo interior to Ro, and will vanish at z=c; while on the boundary of R it will be of constant modulus unity. Thus if it be plotted upon a plane of f the boundary of R will become a circle of radius unit with centre at #=0, this latter point corresponding to z=c. K closed path within Ro, passing once round z =c, will lead to a closed ath passing once about {=o. Thus every point of the interior of will give rise to one point of the interior of the circle. The converse is also true, but is more difficult to prove; in fact, the differential coefficient dr/dz does not vanish for any point interior to R. This being assumed, we obtain a conformal representation of the interior of the region R upon the interior of a circle, in which the arbitrary interior point c of R corresponds to the centre of the circle, and, by utilizing the arbitrary constant arising in determining the function Q, an arbitrary point of the boundary of R corresponds to an arbitrary point of the circumference of the circle. There thus arises the problem of the determination of a real monogenic potential function, single valued and finite within a given arbitrary region, with an assigned continuous value at all points of the boundary of the region. When the region is circular this problem is solved by the integral #jud--'.fUdo previously given. . When the region is bounded by the outermost portions of the circumferences of two overlapping circles, it can hence proved that the problem also has a solution; more generally, consider a finite simply connected region, whose boundary we suppose to consist of a single closed path in the sense previously explained, ABCD: joining A to C by two non-intersecting paths AEC, AFC lying within the region, so that the original region # be supposed to be generated by the overlapping regions AECD, CFAB, of which the common part is AECF; sup now the problem of determining d # valued finite monogenic potential function for the region AECD with a given continuous boundary value can be solved, and also the same problem for the region CFAB; then it can be shown that the same problem can be solved for the original area. Taking undeed the values assigned for the original perimeter ABCD, assume arbitrarily values for the path AEC, continuous with one another and with the values at A and C; then determine the potential function for the interior of AECD; this will prescribe values for the path CFA which will be continuous at A and C with the values originally pro for ABC; we can then determine a function for the interior of CFAB with the boundary values so prescribed. This in its turn will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which this assumes along CFA we can then again determine a new function for the interior of CFAB. And so on. It can be shown that these functions, so alternately determined, have a limit representin such a potential function as is desired for the interior of the origina region ABCD. There cannot be two functions with the given perimeter values, since their difference would a monogenic potential function with boundary value zero, which can easily be shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem. A particular case of the problem is that of the conforcial £; sentation of the interior ''' glosed polygon upon the upper half of the £ of a complex variable t. It can be shown without much difficulty that if a, b, c, ... be real values of t, and a, 6, y, ... be rs real numbers, whose sum is n-2, the integral 2= f(t-a)*-*(t–b)8-'...dt, as t describes the real axis, describes in the plane of z a polygon of n sides with internal angles equal to ar, 6.x, ..., and, a proper sign being given to the integral, points of the upper half of the plane of t give rise to interior points of the polygon. erein the points a, b, ... of the real axis give rise to the corners of the polygon; the condition Za =n-2 ensures merely that the point t = do does not, correspond to a corner; if this condition be not regarded, an additional corner and side is introduced in the polygon. Conversely it can be shown that the conformal representation of a polygon upon the half plane can be effected in this way; for a £ of given position of more than three sides it is necessary for this to determine the positions of all but three of a, b, c, ...; three of them may always be supposed to be at arbitrary positions, such as t-o, t = 1, 1 = 20. As an illustration consider in the plane of z, = x + iy, the portion of the imaginary axis from the origin to z =ih, where h is itive and less than unity; let C this point z = ih; let BA be of length unity along the positive real axis, B being the origin and A the int z = 1; let DE be of length unity along the negative real axis, being also the origin and E the point z=-1; let EFA be a semicircle of radius unity, F being the point x=i. If we put t=[(*+h')/(f+h')]", with t-1 when z=1, the function is single valued within the semicircle, in the plane of z, which is slit along the imaginary axis from the origin to a = th; if we plot the value of upon another plane, as z describes the continuous curve ABCDE, ! will describe the real axis from t=1 to #5-1, the point C giving #=o, and the points B, D giving the points t= .# Near z =o takes each real value once as a passes along the perimeter of the triangle ODE, being as can be shown respectively co, 1, o, -1 at 0, D. H. E., and takes every complex value of imaginary part positive once in the interior of this triangle. This leads to QQ u = tiJ. (*-1)-'du in accordance with the general theory. It can be deduced that r =s* represents the triangle ODH on the upper half £" of r, and t = (1-r") represents similarly the triangle OBD. § 16. Multiple valued Functions. Algebraic Functions.—The explanations and definitions of a monogenic function hitherto given have been framed for the most part with a view to single valued functions. But starting from a power series, say in z-c, which represents a single value at all points of its circle of convergence, suppose that, by means of a derived series in z-c, where c’ is interior to the circle of convergence, we can continue the function beyond this, and then by means of a series derived from the first derived series we can make a further continuation, and so on; it may well be that when, after a closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the original value. One example is the case 2', for which two values exist for any value of z; another is the generalized logarithm X(s), for which there is an infinite number of values. In such cases, as before, the region of existence of the function consists of all points which can be reached by such continuations with power series, and the singular points, which are the limiting points of the point-aggregate constituting the region of existence, are those points in whose neighbourhood the radii of convergence of derived series have zero for limit. In this description-the point z = to does not occupy an exceptional position, a power series in 2-c being transformed to a series in 1/2 when z is near enough to c by means of 2-c =c(1-cz")[1-(1-cz")]", and a series in 1/2 to a series in z-c, when z is near enough to c, by - z - -1 means of #-: (#) The commonest case of the occurrence of multiple valued functions is that in which the functions satisfies an algebraic equation f(s.3) = pos"+pus"+ ... +p =o, wherein po, p1, ... pn are integral polynomials in 2: Assuming f(s;z) incapable of being written as a product of polynomials rational in s and 2, and excepting values of a for which the polynomial coefficient of s” vanishes, as also the values of 2 for which beside f(s.2)=o we have also of(5,2)/3s =o, and also in general the point z=zo, the roots of this equation about any point * = c are given by n power series in 3-c.. About a finite point g=c for which the equation of(s;z)/3s =o is satisfied by one or more of the roots s of f(s,t)=0, the n roots break up into a certain number of cycles, the r roots of a cycle being given by a set of power series in one another by replacing (2-c)"/" by w(z-c)"/", where w, equal to exp (2 rih/r), is one of the rth roots of unity. Putting then z-c = r we may say that the r roots of a cycle are given by a single power series in t, an increase of 2r in the phase of t giving an increase of 2*r in the phase of z–c. This single series in t, giving the values of s belonging to one cycle in the neighbourhood of z = c when the phase of 2-c varies through 2rr, is to be looked upon as defining a single place among the aggregate of values of z and s which £5 = 0; two such places may be at the same point (z=c, s =d), without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of s, z =c, there are n places for which the neighbouring values of s are given by n power series in z-c; for a value of s for which of(s,z)/ds =o there are less than n places. Similar remarks hold for the neighbourhood of 2 = 20 ; there may be n places whose neighbourhood is given by n power series in 2-1 or fewer, one of these being associated with a series in t, where t = (−1)"; the sum of the values of r which thus arise is always n. In general, then, we may say, with t of one of the forms (5-c), (2-6)'', 2', (r)'', that the neighbourhood of any place (c,d) for which f(c,d) =o is given by a pair of expressions * = c-HP(t), s =d-i-Q(t), where P(t) is a (particular case of a) power series vanishing for t=o, and Q(t) is a power series vanishing for t =o, and t vanishes at (c,d), the £ z-c being replaced by s" when c is infinite, and similarly the expression s-d by s- when d is infinite. The last case arises when we consider the finite values of 2 for which the polynomial coefficient of s” vanishes. Of such a pair of expressions we may obtain a continuation by writing t = to + Mir-HAzr"+ ...., where r is a new variable and Al is not zero; in particular for an ordinary finite place this equation simply becomes t = to +r. . It can be shown that all the pairs of power series z= c-H P(t), # which are necessary to represent all pairs of values of *, * satisfying the equation f(s,z) =o can be obtained from one of them by this process of continuation, a fact which we express by saying that the equation f(s,z) =o defines a monogenic algebraic construct. With less accuracy we may say that an irreducible £ic equation f(s,z) =o determines a single monogenic function s Ol 2. Any rational function of sands, where f(x,z)=o, may be considered in the neighbourhood of any place (c,d) by substituting therein z = c +P(t), s =d-HQ(t); the result is necessarily of the form t-H(t). where H(t) is a power series in t not vanishing for t =o and m is an integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative, = -u, the function is infinite to order u at the place... More generally, if A be an arbitrary constant, and, near (c,d), R(s;z)-A is of the form t-H(t), where m is positive, we say that R(s,z) becomes m times equal to A at the place; if R(s,z) is infinite of order u at the place, so also is R(s,z)-A. It can be shown that the sum of the values of m at all the places, including the places z=00, where R(s,z) vanishes, which we call the number of zeros of R(s,z) on the algebraic construct, is finite, and equal to the sum of the values of u where R (s.2) is infinite, and more generally equal to the sum of the values of m where #: this we express by saying that a rational function R(s,z) takes any value '. co) the same number of times or the algebraic construct; this number is called the order of the rational function. That the total number of zeros of R (s,z) is finite is at once obvious, these values being obtainable by rational elimination of s between f(5,2) =0, R(s,z) =o., That the number is equal to the total number of infinities is best deduced by means of a theorem which is also of more general utility. Let R(s,z) be any rational function of s, z, which are connected by f(s,z) = or about any place (c,d) for which z=c-i-P(t), s =d-HQ(t), expand the product |