+ ܝܗ We have seen above that all single valued doubly periodic mero- | paths Beginning and ending at the arbitrary point a each enclosing two variables s,s connected by an equation of the form 5=4* one or more of the exceptional points, these closed paths being A:+B. Taking account of the relation connecting these variables s,: chosen, when |(3) is not a single valued function, so that the final with the argument of the doubly periodic functions (which was above value of f(x) at o is equal to its initial value. It is necessary for denoted by 2), it can then easily be seen that the theorem now proved the statement that this condition may be capable of being is a generalization of the theorem proved previously establishing for satisfied. a doubly periodic function a definite order. There exists a generalization of another theorem also proved above for doubly periodic functions, namely, that the sum of the values of the argument in one For instance, the integral s*r'dz is liable to an additive indeterparallelogram of periods for which a doubly periodic function takes minateness equal to the value obtained by a closed path about s=0, a given value is independent of that value this generalization, which is equal to ari; if we put *= f'side and consider : as a known as Abel's Theorem, is given § 17 below. $ 17. Inlegrals of Algebraic Functions.-In treatises on Integral function of u, then we must regard this function as unaffected by the addition of 2ri to its argument w; we know in fact that Calculus it is proved that if R(z) denote any rational function, 2= exp (w) and is a single valued function of u, with the period 2ri. an indefinite integral [R(z)da can be evaluated in terms of Or again the integral . (1+2+)-'ds is liable to an additive indetermetrical functions. In generalization of this it was long ago minateness equal to the value obtained by a closed path about discovered that if pe=aza +bz+c and R(sx) be any rational either of the points.8- *;; thus if we put u S*(1+3)-'da , the function of s, z any integral / R(S.2) dz can be evaluated in terms function of u is periodic with period #, this being the function of rational functions of s, 2 and logarithms of such functions; tan (w). Next we take the integral »= S(1-44)-ids, agreeing that the simplest case is sods or (ax+68+c) odz. More generally the upper and lower limits refer not only to definite values of 2, but if f(s, z)=o be such a relation connecting s, that when 0 is an to dehnite values of s each associated with a definite determination appropriate rational function s and a both s and z are rationally of the sign of the associated radical (1-3). We suppose its. expressible, in virtue of f(s,z)=0 in terms of 0, the integral 1-2 each to have phase zero for 2; then a single closed circuit (R(s,z)dz is reducible to a form (H(0)do, where H() is rational of 2--I will lead back to :=o with (1-6)} = -1; the additive in 0, and can therefore also be evaluated by rational functions restores the initial value of the subject of integration, may be indeterminateness of the integral, obtained by a closed path which and logarithms of rational functions of s and ... It was natural obtained by a closed circuit containing both the points = 1 in its to inquire whether a similar theorem holds for integrals interior: this gives, since the integral taken about a vanishing (R(3,3)dz wherein s> is a cubic polynomial in 2. The answer is circle whose centre is either of the points := *I has ultimately the value zero, the sum in the negative. For instance, no one of the three integrals S S Sotes So got Linea +S. - It Still can be expressed by rational and logarithms of rational functions where, in each case, (1-7°)) is real and positive; that is, it gives of.s and 2; but it can be shown that every integral /R(s,z)dz So can be expressed by means of integrals of these three types TI-?)] together with rational and logarithms of rational functions of or 27. Thus the additive indeterminateness of the integral is of the s and 2 (see below under $ 20, Elliptic Integrals). A similar form 2km, where k is an integer, and the function z of 4, which is theorem is true when so = quartic polynomial in z; in fact when sin (u), bas 2r for period. Take now the case sé =A(2-a)(2-1)(3-c)(z-d), putting y=s(2-a)], x=(2-a), -=S%) 712-a)(3-653–c)(3-d)) we obtain ya=cubic polynomial in %. Much less is the theorem true when the fundamental relation f(5,2)=0 is of more general adopting a definite determination for the phase of each of the type. There exists then, however, a very general theorem, factors 2-e, sb, 3-6, -d at the arbitrary point w, and supposing known as Abel's Theorem, which may be enunciated as follows: the upper limit to refer, not only to a definite value of s, but also Beside the rational function R(s, 2) occurring in the integral From describe a closed loop about the point s=, consisting, to a definite determination of the radical under the sign of integration. (R(8,3)dz, consider another rational function H(s,x); let suppose, of a straight path from a to a, followed by a vanishing (@:),... (am) denote the places of the construct associated circle whose centre is at a, completed by the straight path from a with the fundamental equation {(s,z)=0, for which H(5,2) is Let similar loops be imagined for each of the points b, c, d, no two of these having a point in common. Let A denote the value equal to one value A, each taken with its proper multiplicity, obtained by the positive circuit of the first loop; this will be in lact and let (61), .. (bm) denote the places for which H(s,z)=B, equal to twice the integral taken from along the straight path where B is another value; then ihe sum of the m integrals to a; for the contribution due to the vanishing circle is ultimately SR(5,2 ) dz is equal to the sum of the coefficients of F-in the the subject of integration. After the circuit about as we string expansions of the function B, C, D denote the values of the integral taken by the loops en closing respectively b, c and d when in each case the initial deterR(S,Bleon (H(6, 7) = 8). mination of the subject of integration is that adopted in calculating A. If then we take a circuit from zo enclosing both e and 6 but where I denotes the generalized logarithmic function, at the not either c or d, the value obtained will be A-B, and on returning various places where the expansion of R(3,2)dz[di contains thus that the integral is subject to an additive indeterminateness negative powers of l. Thiş fact may be obtained at once from equal to any one of the six differences such as A-B. Of these the equation there are only two linearly independent; for clearly only A-B, A-C, A-D are linearly independent, and in fact, as we see by =9, taking a closed circuit' enclosing all of a, b, c, d, we have A-B* wherein w is a constant. (For illustrations see below, under about which the subject of integration suffers a change of sign, and a C-D=0; for there is no other point in the plane beside a, b, c, d $ 20, Elliptic Integrals.) circuit enclosing all of a, b, c, d may by putting = 1/5 be reduced to a $ 18. Indeterminateness of Algebraic Integrals.-The theorem circuit about <=0 about which the value of the integral is zero. that the integrals:/(z)dz is independent of the path from a to ated sign of the radical, when we start with a definite determination s, holds only on the hypothesis that any two such paths are of the subject of integration, is thus seen to be of the form equivalent, that is, taken together from the complete boundary A-B is independent of the position of so , being obtainable by a single 10+m(A-B)+n(A-C), where m and n are integers. The value of of a region of the plane within which f(z) is finite and single closed positive circuit about a and bonly; it is thus equal to twice the valued, besides being differentiable. Suppose that these con- integral taken once from a to b, with a proper initial determination ditions fail only at a finite number of isolated points in the finite of the radical under the sign of integration. Similar remarks to the part of the plane. Then any path from a to z is equivalent, lunction ofz; in any such case H(E) is a rational supction of sand a in the sense explained, to any other path together with closed, quantity s connected therewith by an irreducible rational algebraic to do. =0; P.-S (*+even) e ES 3 equation f(s, 3) =0. Such an integral SK(8, s)ds is called an Abelian ( Integral. Js, as we easily see. If then we have any elliptic integral 19. Reversion of an Algebraic Integral.-In a limited number of having algebraic infinities we can, by subtraction from it of an cases the equation u= S; H(o)ds , in which H(2) is an algebraicfunction appropriate sum of constant multiples of Jo. Jo, J. and their differof s, defines s as a single valued function of w.. Several cases of this without algebraic infinities. But, in fact, if J, J' denote any two have been mentioned in the previous section; from what was of the three integrals J!, J., Jo, there exists an equation AJ+BJ'+ previously proved under $.14., Doubly Periodic Functions, it appears Cfs-ds=rational function of 3, s, where A,B,C are properly chosen that it is necessary for this that the integral should have at most constants. For the rational function two linearly independent additive constants of indeterminateness; for instance, for an integral stoves --S 1(-a) (8-(-c)(8-0) (-c)(-81-tda, is at once found to become infinite for (20. so), not for (20.-se), its infinite part for the first point being 25/(-2), and to become there are three such constants, of the form A-B, A-C, A-D, infinite for s infinitely large, and one sign of s only when these are which are not connected by any linear equation with integral co separable, its infinite part there being 22 V a. or avdiva when &o=0. efficients, and z is not a single valued function of u. It does not become infinite for any other pair (s, s) satisfying the § 20. Elliptic Inlegrals.-An integral of the form SR(3,s)dz, relation se=f(s); this is in accordance with the easily verified where s denotes the square root of a quartic polynomial in s, equation which may reduce to a cubic polynomial, and R denotes a st*+s76.-J.+Je+(948 +2014) Srational function of z and s, is called an elliplic integral. To each value of x belong two values of s, of opposite sign; start and there exists the analogous equation ing, for some particular value of z, with a definite one of these two values, the sign to be attached to s for any other value of a will be iotsva–Je+Je+(+20,0) Sdetermined by the path of integration for z. When 3 is in the neigh. Consider now the integral. bourhood of any finite value o for which the radical ş is not zero, if we put 8-= h, we can find s-50- a power series in 1, say Pseso +2(1); when z is in the neighbourhood of a value, a, for which s vanishes, if we put s=a+f, we shall obtain s=iQ(t), where Q() is a this is at once found to be infinite, for finite values of s, only for power series in 2; when 2 is very large and so is a quartic polynomial (20,5 in s, if we put it, we shall find EQU); when : is very large of s only when these are separable, its infinite part being -log, its infinite part being log (-20), and for 2-0, for one sign and' sa is a cubic polynomial in s, if we put =p, we shall find that is - log : when 650, and - log (sl) when ap=0. g="Q(). By means of substitutions of these forms the character (0)=0, the integral And, if of the integral (R(2, s)ds may be investigated for any position of :: in any case it takes a form SIHI+Komit...+PPR+S4+...jdi involving only a finite number of negative powers of t in the subject of integration. Consider first the particular case folds; it is easily is infinite at z=0, s=0 with an infinite part log I, that is log (3-0), seen that neither for any finite nor for infinite values of a can negative is not infinite for any other finite value of 2, and is infinite like P for powers of 1 enter; the integral is everywhere finite, and is said to be 2=. An integral possessing such logarithmic infinities is said of the first kind; it can, moreover, be shown without difficulty that to be of the third kind. no integral [R(2, 5)ds, save a constant multiple of s'dz, has this Hence it appears that any elliptic integral, by subtraction from property. Consider next, s being of the form oz +40120+ it of an appropriate sum formed with constant multiples of the wherein a, may be zero, the integral S(102° +2012)s-idz; for any finito but for infinite values of z its value is of the form Arl+Q00), where with constant multiples of integrals such as P or P, with constant value nantihis integral is easily proved to be everywhere finite; integral J. and the rational functions of the form (-) Q() is a power series; denoting by vas a particular square root of a multiples of the integral uruz, and with rational functions, when ay is not zero, the integral becomes infinite for := co for both can be reduced to an integral H becoming infinite only for :=.. signs of $, the value of A being + Veo or - Voo according as s is for one sign of s only when these are separable, its infinite part being 700.3(++ ...) or is the negative of this; hence the integral of the form A logo, that is. A log 3 or A log (zl). Such an integral H=/R(2,5)dz docs not exist, however, as we at once find by writing 902? +20,8 J.-S (603 +v0o) ds becomes infinite when 2 is infinite, for and examining the forms possible for these in order that the integral R(2,3)=P(2)+sQ(), where P(?), () are rational functions of the former sign of s, its infinite term being av do.rl or 2v 20.2, may have only the specified infinity. An analogous theorem holds but does not become infinite for infinite for the other sign of s. for rational functions of z and s; there exists no rational function When Q. - the signs of s for 2 = are not separated, being obtained which is finite for finite values of 2 and is infinite only for s=0 one from the other by a circuit of s about an infinitely large circle, for one sign of s and to the first order only: but there exists a and the form obtained represents an integral becoming infinite as rational function infinite in all to the first order for each of two or before for z=-, its infinite part being 2vator 2val.V2. Similarly more pairs (2, 3), however they may be situated, or infinite to the il zo be any finite value of which is not a root of the polynomial second order for an arbitrary pair (s,s); and any rational function (z) to which sis equal, and so denotes a particular one of the deter. may be formed by a sum of constant multiples of functions such as minations of s for 2=2, the integral J.-S{ s: +1(3-2)}"(20) and their differential cocfficients. wherein f'(x) =d[(*)/ds, becomes infinite for 2= 20. s=so, but not for The consideration of elliptic integrals is therefore reducible to s=20,s= -50, its infinite term in the former case being the negative of that of the three 250/3-2);, For no other finite or infinite value of : is the integral infinite. ! :-0 be a root of f(2), in which case the corresponding 2.31 +2012 . sveDEMO +zvao ves value of s is zero, dz respectively of the first, second and third kind. Now the equation se = 607* t... =00(2-0)(2-6)(2_)(2-x), by putting becomes infinite for z=0, its infinite part being, il s-0=R, equal to y=2s(2-6)-700(8-) (0--4) (0-x)]-')}t; and this integral is not elsewhere infinite. In each of these cases, of the integrals J, J2, Js, the subject of integration *===o+;6=+=tów has been chosen so that when the integral is written ncar its point of is at once reduced to the form y? - 4** – 847-82=4(x-2)(x-c(r-es). infinity in the form Ar+Br?+Q0di, the coefficient B is zero, say; and these equations enable us to express s and 2 rationally so that the infinity is of algebraic kind, and so that, when there are in terms of x and y. It is therefore sufhcient to consider three two signs distinguishable for the critical value of 2, the integral elliptic integrals becomes infinite for only one of these. An integral having only algebraic infinities, for hnite or infinite values of z, is called an *-*o ay integral of the second kind, and it appears that such an integral of these consider the first, putting can be formed with only one such infinity, that is, for an infinity prodr. arising only for one particular, and arbitrary, pair of values (s, 2) satisfying the equation så = {(z), this infinity being of the first order. where the limits involve not only a value for x, but a definite sign A function having an algebraic infinity of the mth order (m> 1). for the radical y. When x is very large, if we put lenni. only for one sign of s when these signs are separable, at (1) =, 21°(1 - teatt-8), we have (3) s-.. (3) 3-6, is given respectively by ( 2.) **)..(*) $ 2 - 20 -Sonn whereby a definite power series in u, valid for sufficiently small value, it appears that these periods are sums of integral multiples of two of u, is found for l, and hence a definite power series for x, of the form which may be taken to be x=u?+ bezu*+... Let this expression be valid for o</ul<R, and the function defined thereby, which has a pole of the second order for w=0, be denoted these quantities cannot therefore have a real ratio, for else, being by $(w). In the range in question it is single valued and satisfies the periods of a monogenic function, they would, as we have previously differential equation seen, be each integral multiples of another period; there would lo'(u)p=414(u)]:84(x)-83; then be a closed path for (x,y), starting from an arbitrary point in terms of it we can write x =$(u), y=-&'(u), and, 6'(u) being an (xo, yo), other than one enclosing two of the points (41,0). 2,0). odd function, the sign attached to y in the original integral for x=0 (3,0), (0,0), which leads back to the initial point (xo,yo), which is is immaterial. Now for any two values u, v in the range in question impossible. On the whole, therefore, it appears that the function consider the function (u) agrees with the function P(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under $14. Doubly Periodic Functions. (u)-(). $ 21. Modular Funcions.-One result of the previous theory it is at once seen, from the differential equation, to be such that OF 10u = oFlow; it is therefore a function of uto; supposing is the remarkable fact that if lu tul<R we inser therefore, putting v=0, that dx (v) where yo= 4(x-2)(x-ez)(x-es), then we have By repetition of this equation we infer that if us. ... Un be any argu 1=($w)-° +E'{[(m+!)wtm'w')--{mwtm'w')-). ments each of which is in absolute value less than R, whose sumisalso and a similar equation for es, where the summation refers to in absolute value less than R, then (unt function of the 2n functions (us), $'(u); and hence, if luf<R, all integer values of m and mother than the one pair m=0, that m' =0. This, with similar results, has led to the consideration of functions of the complex ratio w'lw. It is easy to see that the series for B(u), u?+E'[(u+mw +m'w)?where H is some rational function of the arguments $(u/n), *'(u/n). (mw + m'w')*), is unaffected by replacing w, w' by two quantities 'n In fact, however, so long as lu/nl< R, each of the functions (un), equal respectively to pw+qw. pw+w, where p. 9 p.d are any $'(un) is single valued and without singularity save for the pole at integers for which pa'-p'9=*1; further it can be proved that ali u=0; and a rational function of single valued functions, each of substitutions with integer coefficients = pw+qw',' ' = p'w+g'w which has no singularities other than poles in a certain region, is wherein pa'-p'q=1, can be built up by repetitions of the two par: also a single valued function without singularities other than poles in ticular substitutions (12=-W, SY =w). (2=w, &' =w+w). Consider this region. We infer, therefore, that the function of u expressed by the function of the ratio wlw expressed by. h=-B(w')/P(*w); H[•C*), « (*) ) is single valued and without singularities other it is at once seen from the properties of the function ®(that by than poles so long as (ul<nR; it agrees with p(u) when lu}<R, and the hence furnishes a continuation of this function over the extended sponding substitutions for h expressed by range lul<n R. Moreover, from the method of its derivation, it l'=1/k, l'=i-h; satisfies the differential equation (&'(u)} = 4[*(u)- 820(u) - 82. This thus, by all the integer substitutions ? = pw+qw', s = p'wto'w', in equation has therefore one solution which is a single valued mono which po' -P'9= 1, the function h can only take one of the six values genic function with no singularities other than poles for any finite k, U/h, 1-h. 17(1-h), h/(k-1), (k-1)/k, which are the roots of an part of the plane, having in particular for uro, a pole of the second equation in 0, order; and the method adopted for obtaining this near u=0 shows that the differential equation has no other such solution. This, (1-0+02)_(1-k+ha)'. however, is not the only solution which is a single valued mero ( 10)?T?(1 – h)?: morphic function, all the functions (u+a), wherein a is arbitrary, the function of t, Eww, expressed by the right side, is thus being such. Taking now any range of values of 1, from u=o; unaltered by every one of the substitutions s'='10's, wherein and putting for any value of u, x=0(u), y=-*'(u), so that P+QT' goed = 478-827-88, we clearly have P. 9, ',¢are integers having po' -p'q=1.. If the imaginary part 6, 0f 1, which we may write raptio, is positive, the imaginary part of 7', which is equal to o(pa-pa)/(p+9p)'+q*0*), is also positive; (sv) suppose o to be positive; it can be shown that the upper half of the conversely if xo = 4(u), Yo = -6'(un) and 5, 7 be any values satisfying infinite plane of the complex variable r can be divided into regions, m=46-85-85, which are sufficiently near respectively to x6, yo, all bounded by arcs of circles (or straight lines), no two of these while o is defined by regions overlapping, such that any substitution of the kind under consideration, i' = (p'+q'r)/(P+qr) leads from an arbitrary points, 0-2= of one of these regions, to a point ' of another; taking r=ptio, one of these regions may be taken to be that for which - p<. then €, 9 are respectively $(v) and -&'(v); for this equation leads p2+o>!, together with the points for which p is negative on the to an expansion for 5-Xo in terms of v=un, and only one such ex curves limiting this region; then every other region is obtained pansion, and this is obtained by the same work as would be necessary from this so-called fundamental region by one and only one of the to expand o(o) when v is near to uo; the function $(u) can therefore substitutions 7=(0'+g'r)!(P+or), and hence by a definite combinabe continued by the help of this equation, from v=uo, provided tion of the substitutions t'=-1/1.7'=1+. Upon the infinite half the lower limit of $-xol necessary for the expansions is not zero plane of 7, the function considered above, in the neighbourhood of any value (xo,yo). In fact the function (u) (B?(jw)+ P(!w)P({w') + Ba(w)| can have only a finite number of poles in any finite part of the plane 2() = 24 PC1w) (1W)(Iw) + P(jw Circles, the lower limit of the radil of convergence of the expansions those for which is any real rational value; the real axis is thus a portion of the finite part of the plane of u which is outside these is a single valued monogenic function, whose only essential singuof o(u) is greater than zero; the same will therefore be the case for the lower limit of the radii (8 - xol necessary for the continuations line over which the function z(1) cannot be continued, having an spoken' of above provided that the values of (5, 7) considered do not essential singularity in every arc of it, however short; in the fundalead to infinitely increasing values of v; there does not exist, how- mental region, (v) has thus only the single essential singularity, ever, any definite point (50,70) in the neighbourhood of which the assigned complex value just once, the relation z(:')=2(1) requiring, integral Sector (6.1) de as can be shown, that ' is of the form (+ar)/(P+97), in which (+0,80) 77 length that the integral can so increase. We infer therefore that a similar behaviour in every other of the regions. The division of if (8.) be any point, where qe = 46-826-83, and obe-defined by the plane into regions is analogous to the division of the plane, ) dx in the case of doubly periodic functions, into parallelograms; in that case we considered only functions without essential singularities, and in each of the regions the function assumed every complex value then £=**) and y=-*'(o). Thus this equation determines (E. n) twice, at least. Putting, as another function of 7. J(1) = 2(v) [z(r) – 1, without ambiguity. In particular the additive indeterminatenesses it can be shown that J(7) = 0 for 1 = exp (1x), that (1) = 1 for 7=i, of the integral obtained by closed circuits of the point of integration these being values of : on the boundary of the fundamental'region; are periods of the function (u); by considerations advanced above I like :(r) it has an essential singularity for 1 =ptio, o=+0. In the of w; int theory of linear differential equations it is important to consider the 7-plane expressed by (p+1) +0=,(-1) + 0 = f, while the two inverse function (), this is infinitely many valued, having a cycle sides of the latter portion, for which is real and>1, correspond of three values for circulation of J about j = 0 (the circuit of this to the lines of the r.plane expressed by p=+1. The line for point leading to a linear substitution for of period 3, such as which , is real, positive and less than unity corresponds to the =-(1+1)-1), having a cycle of two values about J=1 (the circuit imaginary axis of the 7-plane, lying in the interior of the sunda. leading to a lincar substitution for 1 of period 2, such as a' =7), mental region. All the values of r = s(0, 0, 0, 1) may then be derived and having a cycle of infinitely many values about J = (the circuit from those belonging to the fundamental region of the 7.plane by leading to a linear substitution for i which is not periodic, such as making a describe a proper succession of circuits about the points p=1+r). These are the only singularities for the function (). =o, N=1; any such circuit subjects 7 to a linear substitution Each of the functions of the subgroup of r considered, and corresponds to a change of 1 (-)}. IJ(+)-1], [- )=213) from a point of the fundamental region to a cortesponding point of one of the other regions. beside many others (see below), is a single valued function of T, § 22. A Property of Integral Functions deduced from the Theory and is expressible without ambiguity in terms of the single values of Modular Functions.-Consider now the function exp(2), function of r, for finite values of 2; for such values of z, exp (2) never vanishes, 7()=exp 11, 11-exp (zirnt)] and it is impossible to assign a closed circuit for 2 in the finite part of the plane of z which will make the function \=exp(z) =exp (02) - E (-1)" exp 1(3m? +m)int). pass through a closed succession of values in the plane of a having 1=0 in its interior; the function s[0,0,0, exp (3)], It should be remarked, however, that 9() is not unaltered by all however z vary in the finite part of the plane, will therefore never the substitutions we have considered; in fact be subjected to those linear substitutions imposed upon (---1)=(-17)\n(1), 7(1+1)=exp(t' iz)n(). The aggregate of the substitutions a' = (p+)/(p+gs), wherein (0,0,0,.) by a circuit of about 1=o; more generally, if p. 2. p. are integers with pa-p'q=1, represents a Group; the o(a) be an integral function of s, never becoming either zero or function }(), unaltered by all these substitutions, is called a Modular unity for finite values of 2, the function 1 =(z), however z vary Function. More generally any function unaltered by all the sub- in the finite part of the plane, will never make, in the plane of X, stitutions of a group of lincar substitutions of its variable is called an a circuit about either 1=o or 1=1, and s(0,0,0,1), that is Automorphic Function. A rational function, of its variable h, of this $10,0,0,4(3)], will be single valued for all finite values of s; character, is the function (1-h+h)'-*(1-h). presenting itself incidentally aboye;. and there are other rational functions with a it will moreover remain finite, and be monogenic. In other similar property, the group of substitutions belonging to any one words, s(0,0,0,0(=)) is also an integral function-whose imaginary of these being, what is a very curious fact, associable with that of part, moreover, by the property of s(0,0,0, X), remains positive the rotaties of one of the regular solids, about an axis through its for all finite values of z. In that case, however, explis(0,0,0,0(2)]} centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already would also be an integral function of 2 with modulus less than discussed; these, as we have seen, enable us to solve the algebraic unity for all finite values of 2. If, however, we describe a circle equation ya = 4x2-828-8s (and in fact many other algebraic equa- of radius R in the : plane, and consider the greatest value of the tions, see below, under $23, Geometrical Applications of Elliptic modulus of an integral function upon this circle, this certainly Functions) in terms of single valued functions * = (x), x=- B': increases indefinitely as R increases. We can infer therefore functions in general: but it can be shown that such functions that an integral function (2) which does nol vanish for any finile necessarily have an infinite number of essential singularities except value of 2, takes the value unity and hence (by considering the for the simplest cases. The modular function J(1), considered above, unaltered by the function A-10(2)) takes every other value for some definite salue group of linear substitutions ;' =(p+g'r)/(p+97), where p.9.0.gf ; or, an integral function for which both the equations are integers with po'-e'q=1, may be taken as the independent $(3) = A,“(z)=B are unsatisfied by definite values of s,does not variable x of a differential equation of the third order, of the form exist, A and B being arbitrary constants. 1- Ha? +62-22 A similar theorem can be proved in regard to the values assumed 2(x-1)2T 2r? 2x(x-1) by the function (2) for points of modulus greater than R, however where s' =ds/dx, &c., of which the dependent variable s is equal to 1. great R may be, also with the help of modular functions. In general A differential equation of this form is satisfied by the quotient of integral function not to assume every complex value an infinite terms it may be stated that it is a very exceptional thing for an two independent integrals of the linear differential equation of the number of times. second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(Q.B.7.x), function sla, B. , ) is a single valued function of 1 = s(0, 0, 0, 1); Another application of modular functions is to prove that the we have in fact = s(}, }, o, J). If we introduce also the function for, putting 1 = (1-1)/(1+i), the values of s' which correspond to the of r given by 2B(°) + P({w) singular points 1 =0, 1, -0 of s(a, ß, 1), though infinite in number, all lic on the circumference of the circle 1:' =1, within which therefore (1w") - P(w) we similarly have q=s(0,0,0,1); this function 1 is a single valued sla, B. 7, 8) is expressible in a form & ant'n. More generally any function of T, which is also a modular function, being unaltered by a group of integral substitutions also of the form o'=('+q7)/(p+97), the points d = 0, 1, 00, is a single valued function of r = 5(0, 0, 0, 1); monogenic function of a which is single valued save for circuits of with po-p'a=!, but with the restriction that p and are even identifying 1 with the square of the modulus in Legendre's form of integers, and therefore p and d' are odd integers. This group is the elliptical integral, we have.r = įK'/K, where thus a subgroup of the general modular group, and is in fact of the kind called a self-conjugate subgroup. As in the general case this K-Svi- f 1 -17. K:=S, V/1-4][4-(1 - 1)a): subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the functions such as 1 (1-1). [1(1-1))", which have only 1 = 0, 1, 00 fundamental region, by a j rticular one of the substitutions of the as singular points, were expressed by Jacobi as power series in greint, subgroup. This fundamental region, putting =ptio, may be and therefore, at least for a limited range of values of o, as single taken to be that given by-1 < 51, 671)' to>(o-)'to>; valued functions of r; it follows by the theorem given that any and is built up of six of the regions which arose for the general product of a root of 4 and a root of 1-1 is a single valued function modular group associated with }()Within this fundamental of 7. More generally the differential equation region, 1 takes every complex value just once, except the values X = 0, 1, 0, which arise only at the angular points r=0,2 =0,7=-1 *(1-x)+r(a+B+1)xlente-aBy=o and the equivalent point i = 1;, these angular points åre essential may be solved by expressing both the independent and dependent singularities for the function (7). For 1(7) as for J(7), the region of variables as single valued functions of a single variable 7, the expres. existence is the upper half plane of ,, there being an essential singu- sion for the independent variable being x=(r). larity in every length of the real axis, however short. If, beside the plane of 1, we take a plane to represent the values of $23. Geometrical Applications of Eliptic Functions.-Consider 4, the function r = s(0, 0, 0, 1) being considered thereon, the values of any irreducible algebraic equation rational in x,y,f(x,y) =0, of belonging to the interior of the fundamental region of the roplane such a form that the equation represents a plane curve of order considered above, will require the consideration of the whole of the A-plane taken once with the exception of the portions of the real n with in(n-3) double points; taking upon this curve 1-3 axis lying between 0 and o and between 1 and to the two arbitrary fixed points, draw through these and the double sides of the first portion corresponding to the circumferences of the points the most general curve of order n - 2; this will intersect f in n(n-3)--(n-3)-(1-3)=3 other points, and will contain that 3+x+x)={m, and hence is HI(n=)/(*1-33)p; so that we homogeneously at least }(n-1)n-fu(n-3)-(n-3)=3 arbi- have another proof of the addition equation for the function B(x). trary constants, and so will be of the form 10+1101+342+ | properties, as for example' those of its inflections, the properties of ... = o, wherein As, de, ... are in general zero. Put now inscribed polygons, of the three kinds of corresponding points, and &=01/4, n=0/and eliminate x,y between these equations and the theory of residuation, are at once obvious. And similar results f(x,y)=0, so obtaining a rational irreducible equation F(8,4)=0, hold for the curve of order n with f(n-3)n double points. representing a further plane curve. To any point (x,y) off will § 24. Integrals of Algebraic Functions in Connexion with the then correspond a definite point (5,9) of F. Theory of Plane Curves. The developments which have been For a general position of (x,y) upon the equations reader to appreciate the vastly more extensive theory similarly explained in connexion with elliptic functions may enable the subject to (2.32 = 0, will have the same number of solutions (33; arising for any algebraical irrationality, f(x,y)=0. if their only solution is x'=x, y'=y, then to any position (.n) of F The algebraical integrals SR(x,y)dx associated with this may as will conversely correspond only one position (x,y) of f. If these before be divided into those of the first kind, which have no inequations have another solution beside (x,y), then any curve finities, those of the second kind, possessing only algebraical infinities, 10+291 +180 =0 which passes (through the double points off and those of the third kind, for which logarithmic infinities enter. and) through the n-2 points of constituted by the fixed n-3 Here there is a certain number, pa greater than unity, of linearly points and a point (x,y), will necessarily, pass through a further independent integrals of the first kind; and this number p is unpoint, say (xo'.yo'), and will have only one further intersection with altered by any birational transformation of the fundamental equation 1; such a curve, with the n-2 assigned points, beside the double 1(x,y)=0; a rational function can be constructed with poles of the points, of f, will be of the form med trivit... =0, where us, Mari.. first order at p+! arbitrary positions (x,y), satisfying f(x,y)=0, are generally zero; considering the curves y twin =0, for variable i, but not with a fewer number unless their positions are chosen one of these passes through a further arbitrary point off, by choosing properly, a property we found for the case p=1; and p is the number properly, and conversely an arbitrary value of I determines a single of linearly independent curves of order 1 – 3 passing through the further point of fi, the co-ordinates of the points off are thus double points of the curve of order » expressed by f(x,y)=0. Again rational functions of a parameter !, which is itself expressible ration- any integral of the second kind can be expressed as a sum of P ally by the co-ordinates of the point; it can be shown algebraically integrals of this kind, with poles of the first order at arbitrary that such a curve has not }(1-3)n but }(n-3)n + double points. positions, together with rational functions and integrals of the first We may therefore assume that to every point of F corresponds kind; and an integral of the second kind can be found with one only one point of f, and there is a birational transformation between pole of the first order of arbitrary position, and an integral of the these curves; the coefficients in this transformation will involve third kind with two logarithmic infinities, also of arbitrary position; rationally the co-ordinates of the n-3 fixed points taken upon f: the corresponding properties for p= are proved above. that is, at the least, by taking these to be consecutive points, will There is, however, a difference of essential kind in regard to the involve the co-ordinates of one point off, and will not be rational inversion of integrals of the first kind; if u=SR(x,y)dx be such an in the coefficients of s unless we can specify a point of whose co- integral, it can be shown, in common with all algebraic integrals ordinates are rational in these. The curve F is intersected by a associated with f(x,y) = 0, to have a linearly independent additive straight line a5+bm.tcrin as many points as the number of constants of indeterminateness; the upper limit of the integral unspecified intersections of f with a$++0=0, that is, 3; or F cannot therefore, as we have shown, be a single valued function will be a cubic curve, without double points. of the value of the integral. The corresponding theorem, if SR:(x,y)dx Such a cubic curve has at least one point of inflection Y, and if a denote one of the integrals of the first kind, is that the pequations variable line YPQ be drawn through Y to cut the curve-again in P and Q, the locus of a point R such that YR is the harmonic mean of determine the rational symmetric functions of the positions (x2.91). SR. (x,yı)dxi +...+sRi(x,yy)dxp =Hi, YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which ... (xp.Yp) as single valued functions of the p variables, ... up: this straight line is Y=0, and the inflexional tangent at Y is 2=0; It is thus necessary to enter into the theory of functions of several the equation of the cubic curve will then be of the form independent variables; the equation f(x,y)=0 is thus not, ZY?=aX' +6X’Z+cXZ2+d2°; in this way, capable of solution by single valued functions of one variable. That solution in fact is to be sought with the help of by putting X equal to AX + 2, that is, choosing a suitable line automorphic functions, which, however, as has been remarked. through Y to be X=0, and choosing À properly, this is reduced to have, for p>1, an infinite number of essential singularities. the form 25. Monogenic Functions of Several Independent Variables.ZY: -4X3-82XZ-83Z, A monogenic function of several independent complex variables of which a representation is given, valid for cyery point, in terms of ", ... is to be regarded as given by an aggregate of power The value of u belonging to any point is definite save for sums of series all obtainable by continuation from any one of them in a integral multiples of the periods of the elliptic functions, being manner analogous to that before explained in the case of one given by independent variable. The singular points, defined as the (a) ZdX-XIZ limiting points of the range over which such continuation is ZY possible, may either be poles, or polar points of indetermination, where (oo) denotes the point of inflection. .or essential singularities, It thus appears that the co-ordinates of any point of a planc curve, A pole is a point (m 2 ... u) in the neighbourhood of which the f, of order 11 with I(n-3)n double points are expr sible as elliptic functions, there being, save for periods, a definite value of the argu- function is expressible as a quotient of converging power series in ment u belonging to every point of the curve. It can then be shown 4-u)...up ; of these the denominator series D must that if a variable curve, ¢, of order m be drawn, passing through vanish at(u.mp, since else the fraction is expressible as a the double points of the curve, the values of the argument u at the remaining intersections of $ with f, have a sum which is unaffected power series and the point is not a singular point, but the namerator by , the periods. In virtue of the birational transformation this be possible to write D=MD., N= MNo, where M is a converging theorem can be deduced from the theorem that if any straight line cut the cubic yo = 4x+-83x -80, in points (-1), (uz), (4x), the sum power series vanishing at ( 22 ) <); and No is a converging power u+uz+u, is zero, or a period; or the general theorem is a corollary series, in (u.—u.... Up-«)), not so vanishing. A polar point from Abel's theorem proved under 17. Integrals of Algebraic of indetermination is a point about which the function can be Functions. To prove the result directly for the cubic we remark expressed as a quotient of two converging power series, both of that the variation of one of the intersections (x,y) of the cubic which vanish at the point. As in such a simple case as (Ax+By)/ and r, is obtained by differentiation of the equation for the three (ax+by), about x=0, y=0, it can be proved that then the function abscissae, namely the equation can be made to approach to any arbitrarily assigned value by F(x)=471-824-go-(mx+n)3=0, making the variables 41,... Up approach tour,.. by a proper and is thus given by path. It is the necessary existence of such polar points of indx determination, which in case p>2 are not merely isolated points, 12t8m +on which renders the theory essentially more difficult than that of F(X) functions of one variable. An essential singularity is any which and the sum of three such fractions as that on the right for the three does not come under one of the two former descriptions and includes roots of F(x) =0 is zero; hence u+uz+us is independent of the very various possibilities. A point at infinity in this theory is one straight line considered; if in particular this become the inflexional for which any one of the variables , ... up is indefinitely great; tangent each of ur, 23. : vanishes. It may be remarked in passing such points are brought under the preceding definitions by means |