F(u,v) = [('(]-(u)—•(v);

it is at once seen, from the differential equation, to be such that OF/du=0F/ov; it is therefore a function of u+v; supposing Ju+v<R we infer therefore, by putting v=o, that

• (u+v) = 1 [ *' (u) = ¿'(' ') ])* −。(u)—•(v).

By repetition of this equation we infer that if u,... u, be any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then (u1 + . . . +u) is a rational '(,); and hence, if u<R,

function of the 2n functions (u),

that

•(u) = H[• (#). • (4)].

where H is some rational function of the arguments (u/n), p'(u/n). | In fact, however, so long as lu/n<R, each of the functions (u/n), '(u/n) is single valued and without singularity save for the pole at u=0; and a rational function of single valued functions, each of which has no singularities other than poles in a certain region, is also a single valued function without singularities other than poles in this region. We infer, therefore, that the function of u expressed by is single valued and without singularities other than poles so long as [u]<nR; it agrees with 4(u) when [u]<R, and hence furnishes a continuation of this function over the extended range u<nR. Moreover, from the method of its derivation, it satisfies the differential equation ['(u)]2 = 4[¢(u)]3 − g(u)-g3. This equation has therefore one solution which is a single valued monogenic function with no singularities other than poles for any finite part of the plane, having in particular for u=0, a pole of the second order; and the method adopted for obtaining this near u=0 shows that the differential equation has no other such solution. This, however, is not the only solution which is a single valued mero morphic function, all the functions (u+a), wherein a is arbitrary, being such. Taking now any range of values of u, from #=0, and putting for any value of u, x=4(u), y='(u), so that y2=4x3-g2x-gs, we clearly have

dx (Tm))

then E, are respectively (v) and (v); for this equation leads to an expansion for -xo in terms of vuo, and only one such expansion, and this is obtained by the same work as would be necessary to expand (v) when v is near to us; the function (u) can therefore be continued by the help of this equation, from v=uo, provided the lower limit of 1-xo necessary for the expansions is not zero in the neighbourhood of any value (xo,yo). In fact the function (u) can have only a finite number of poles in any finite part of the plane of u; each of these can be surrounded by a small circle, and in the portion of the finite part of the plane of u which is outside these circles, the lower limit of the radii of convergence of the expansions of (u) is greater than zero; the same will therefore be the case for the lower limit of the radii -xol necessary for the continuations spoken of above provided that the values of (, ) considered do not lead to infinitely increasing values of v; there does not exist, how ever, any definite point (0,7%) in the neighbourhood of which the integral de increases indefinitely, it is only by a path of infinite

length that the integral can so increase. We infer therefore that if (7) be any point, where n2 = 4§3 — gz§—gs, and v be defined by (*) dx ($,7) y'

then (v) and = '(v). Thus this equation determines (E, n) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function (u); by considerations advanced above

it appears that these periods are sums of integral multiples of two which may be taken to be

these quantities cannot therefore have a real ratio, for else, being periods of a monogenic function, they would, as we have previously seen, be each integral multiples of another period; there would then be a closed path for (x,y), starting from an arbitrary point (x,y), other than one enclosing two of the points (e,0), (2,0), (e1,0), (∞, ∞), which leads back to the initial point (xo, yo), which is impossible. On the whole, therefore, it appears that the function (u) agrees with the function (u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14, Doubly Periodic Functions.

§ 21. Modular Functions.-One result of the previous theory is the remarkable fact that if

y

where y=4(x-e1)(x—er)(x—es), then we have

e1 =(}w)2+Σ'{[(m+})w+m'w']~2-[mw+m'w']~°}, and a similar equation for es, where the summation refers to all integer values of m and m' other than the one pair m=0, m'o. This, with similar results, has led to the consideration of functions of the complex ratio w'/w.

It is easy to see that the series for B(u), u2+E'[(u+mw+m'w') ?— (mw+m'w')], is unaffected by replacing w, w' by two quantities . equal respectively to pw+qw'. p'w+g'w', where p, q, p'q' are any integers for which pq'-p'q=1; further it can be proved that all substitutions with integer coefficients = pw+qw', !! = p'w+q'w', wherein pq'-p'q=1, can be built up by repetitions of the two particular substitutions (=-w', '=w), (!=w, N'=w+w'). Consider the function of the ratio w/w expressed by. h = P({w')/B(w);

it is at once seen from the properties of the function B(u) that by the two particular substitutions referred to we obtain the corre sponding substitutions for h expressed by k' = 1/h, h'=1-h;

thus, by all the integer substitutions N=pw+qw', s= =p'w+q'w', in which pq'-p'q=1, the function h can only take one of the six values h, 1/h, 1—h. 1/(1 − h), h/(h−1), (h−1)/h, which are the roots of an equation in @,

wherein

the function of T, ='w, expressed by the right side, is thus unaltered by every one of the substitutions r'= p+qr P. 9, p', q' are integers having pq'—p'q=1. If the imaginary part , of 7, which we may write rptio, is positive, the imaginary part of r', which is equal to o(pq' - p′q) /\ ( p + qp)2+gol, is also positive; suppose to be positive; it can be shown that the upper half of the infinite plane of the complex variable r can be divided into regions, all bounded by arcs of circles (or straight lines), no two of these regions overlapping, such that any substitution of the kind under consideration, r' = (p' +qr) / (p+qr) leads from an arbitrary point r, of one of these regions, to a point r' of another; taking p+io, one of these regions may be taken to be that for which =}<p<}, p+o>1, together with the points for which p is negative on the curves limiting this region; then every other region is obtained from this so-called fundamental region by one and only one of the substitutions (p'+g'1)/(p+qr), and hence by a definite combination of the substitutions T' = −1/T, T′ = 1+r. Úpon the infinite half plane of 7, the function considered above,

[B2(}w)+P( }w)B( }w′)+V2(}w')]

2(+) === B2 (w) B2 ( j w ) [ B ( } w ) + B( {w T

is a single valued monogenic function, whose only essential singu. those for which r' is any real rational value; the real axis is thus a larities are the points r' (p+q'r)/(p+qr) for which, namely line over which the function z(7) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, (r) has thus only the single essential singularity, assigned complex value just once, the relation (r)=2(1) requiring, r=ptio, where ; in this fundamental region 2(7) takes any as can be shown, that is of the form (p'+g'r)/(p+qr), in which P. q. p, q' are integers with pq'-p'q=1; the function 2(r) has thus a similar behaviour in every other of the regions. The division of the plane into regions is analogous to the division of the plane, 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 twice, at least. Putting, as another function of r. J(r) = 2(7) [2(7) — 1], it can be shown that J(r)=0 for exp (37), that (r) = 1 for r=i, these being values of on the boundary of the fundamental region; like 2(r) it has an essential singularity for r=p+io, o = +∞. In the

theory of linear differential equations it is important to consider the inverse function (J), this is infinitely many valued, having a cycle of three values for circulation of J about J=0 (the circuit of this point leading to a linear substitution for r of period 3, such as '-(1+1)), having a cycle of two values about J1 (the circuit leading to a linear substitution for r of period 2, such as r'), and having a cycle of infinitely many values about J= (the circuit leading to a linear substitution for which is not periodic, such as 1+). These are the only singularities for the function r(J). Each of the functions

[J(+)], [J(+)-1], [-(1)+ 2 B ( )w'))2. -(1)w')) beside many others (see below), is a single valued function of r, and is expressible without ambiguity in terms of the single valued function of T.

πτ
7(7)=exp (117) 11, [1-exp (2ixnr)],

=exp

1

DO

Σ (-1) exp [(3m2+m)ixr).

It should be remarked, however, that (7) is not unaltered by all the substitutions we have considered; in fact

n(-r ̄1) = (-ir)in(r), n(1+r)=exp(11⁄2 iz)n(1).

The aggregate of the substitutions r'=(p+qr)/(p+qr), wherein P. q. p. are integers with pq-p'q=1, represents a Group; the function (), unaltered by all these substitutions, is called a Modular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called an Automorphic Function. A rational function, of its variable h, of this character, is the function (1-h+h)-(1-h) presenting itself incidentally aboye; and there are other rational functions with a similar property, the group of substitutions belonging to any one of these being, what is a very curious fact, associable with that of the rotatits of one of the regular solids, about an axis through its centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already discussed; these, as we have seen, enable us to solve the algebraic equation y=4x-gx-gs (and in fact many other algebraic equations, see below, under $23, Geometrical Applications of Elliptic Functions) in terms of single valued functions x = P(u), y=- B'(u). A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.

The modular function J(r), considered above, unaltered by the group of linear substitutions = (p'+g'r)/(p+qr), where p, q, P', q' are integers with pq-p'q-1, may be taken as the independent variable x of a differential equation of the third order, of the form I-a? 1- B2, – =-3(-)--+-- §2 + a2 + b2 - ‚2 - 1 2(x-1)+ 2x(x-1) 2x2 where s'ds/dx, &c., of which the dependent variable s is equal to 7. A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(a,B,y,x). we have in fact 7=s(}, }, 0, J). If we introduce also the function of r given by

T

2B({w')+P({w) λ= B({w')−P(}w)'

we similarly have 7=s(0, 0, 0, X); this function A is a single valued function of r, which is also a modular function, being unaltered by a group of integral substitutions also of the form (p+qr)/(p+q), with pq'-p'q=1, but with the restriction that p' and 9 are even integers, and therefore p and q' are odd integers. This group is 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 subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the fundamental region, by a 1 rticular one of the substitutions of the subgroup. This fundamental region, putting rptio, may be taken to be that given by-1<p<I, (p+1)2+o2 > }. (p− } )2 +o>, and is built up of six of the regions which arose for the general modular group associated with J(7). Within this fundamental region, A takes every complex value just once, except the values which arise only at the angular points r =0, T=∞,T=-1 and the equivalent point =1; these angular points are essential singularities for the function (r). For X(7) as for J(7), the region of existence is the upper half plane of 7, there being an essential singularity in every length of the real axis, however short.

λ=0, 1, ∞,

If, beside the plane of r, we take a plane to represent the values of A, the functions(0, 0, 0, λ) being considered thereon, the values of belonging to the interior of the fundamental region of the 7-plane considered above, will require the consideration of the whole of the A-plane taken once with the exception of the portions of the real axis lying between- and o and between 1 and +, the two sides of the first portion corresponding to the circumferences of the

7-plane expressed by (p+1)2+o2 = }, (p−})2 + o2 = 1, while the two sides of the latter portion, for which A is real and >1, correspond to the lines of the 7-plane expressed by p=1. The line for which is real, positive and less than unity corresponds to the imaginary axis of the 7-plane, lying in the interior of the funda mental region. All the values of r = 5(0, 0, 0, λ) may then be derived from those belonging to the fundamental region of the r-plane by making à describe a proper succession of circuits about the points A=o, X=1; any such circuit subjects to a linear substitution of the subgroup of considered, and corresponds to a change of r from a point of the fundamental region to a corresponding point of one of the other regions.

§ 22. A Properly of Integral Functions deduced from the Theory of Modular Functions.-Consider now the function exp(2), for finite values of z; for such values of z, exp (2) never vanishes, and it is impossible to assign a closed circuit for z in the finite part of the plane of z which will make the function λ= exp(2) pass through a closed succession of values in the plane of X having λ=o in its interior; the function s[0,0,0, exp (3)], however z vary in the finite part of the plane, will therefore never be subjected to those linear substitutions imposed upon (0,0,0,X) by a circuit of A about A=0; more generally, if (2) be an integral function of s, never becoming either zero or unity for finite values of 2, the function λ=4(2), however z vary in the finite part of the plane, will never make, in the plane of λ, a circuit about either Ao or λ=1, and s(0,0,0,X), that is [0,0,0,(z)], will be single valued for all finite values of ; it will moreover remain finite, and be monogenic. In other words, s[0,0,0,0(2)] is also an integral function-whose imaginary part, moreover, by the property of s(0,0,0, λ), remains positive for all finite values of z. In that case, however, explis[0,0,0,0(2)]} would also be an integral function of z with modulus less than unity for all finite values of z. If, however, we describe a circle of radius R in the z plane, and consider the greatest value of the modulus of an integral function upon this circle, this certainly We can infer therefore increases indefinitely as R increases. that an integral function (2) which does not vanish for any finile value of z, takes the value unity and hence (by considering the function A-1(z)) takes every other value for some definite value of z; or, an integral function for which both the equations (z) = A, ø(z) = B are unsatisfied by definite values of z, does not exist, A and B being arbitrary constants.

A similar theorem can be proved in regard to the values assumed by the function (2) for points z of modulus greater than R, however great R may be, also with the help of modular functions. In general integral function not to assume every complex value an infinite terms it may be stated that it is a very exceptional thing for an

number of times.

function s(a, B, Y, A) is a single valued function of rs(0, 0, 0, 1); Another application of modular functions is to prove that the for, putting (r-i)/(1+i), the values of 'which correspond to the singular points λ=0, 1, ∞ of s(a, B, y, A), though infinite in number, all lie on the circumference of the circle [r'] = 1, within which therefore s(a, B, v, x) is expressible in a form ar'". More generally any monogenic function of A which is single valued save for circuits of the points λ=0, 1, ∞, is a single valued function of S(0, 0, 0, X). Identifying A with the square of the modulus in Legendre's form of the elliptical integral, we have.r = iK'/K, where dt

functions such as λa, (1-x)'. [X(1−X)]', which have only A=0, 1, as singular points, were expressed by Jacobi as power series in geist and therefore, at least for a limited range of values of r, as single valued functions of r; it follows by the theorem given that any product of a root of λ and a root of 1-A is a single valued function of 7. More generally the differential equation day ,dy

=0 x(1-x)+[r(a+B+ 1 ) x ) = x2 - aßy = (

may be solved by expressing both the independent and dependent variables as single valued functions of a single variable r, the expres sion for the independent variable being x=λ(r).

$23. Geometrical Applications of Elliptic Functions.-Consider any irreducible algebraic equation rational in x,y,f(x,y) =0, of such a form that the equation represents a plane curve of order n with n(n-3) double points; taking upon this curve -3 arbitrary fixed points, draw through these and the double points the most general curve of order n-2; this will intersect

fin n(n-2)-n(n−3)—(n−3)=3 other points, and will contain homogeneously at least (n-1)n-} n (n−3) − (n − 3)=3 arbitrary constants, and so will be of the form λ+1+22+ = 0, wherein As, As, · ... are in general zero. Put now =1/6, n=/ and eliminate x,y between these equations and f(x,y) =o, so obtaining a rational irreducible equation F(n) =0, representing a further plane curve. To any point (x,y) of ƒ will then correspond a definite point (§,7) of F.

For a general position of (x,y) upon f the equations •1(x',y′)/4(x',3'), = 41(x,y)/o(x,y), dz(x′,y')/o(x′,y') = •z(x,y)/p(x,y). subject to f(x,y) =0, will have the same number of solutions (x,y); if their only solution is x'=x, y=y, then to any position (7) of F will conversely correspond only one position (x,y) of f. If these equations have another solution beside (x,y), then any curve λ$+λ11+λ¿$20 which passes (through the double points of f and) through the n-2 points of ƒ constituted by the fixed n-3 points and a point (xo,yo), will necessarily pass through a further point, say (xo'yo'), and will have only one further intersection with f; such a curve, with the n-2 assigned points, beside the double points, of f, will be of the form we tuikt... =0, where μ2, pay... are generally zero; considering the curves +0, for variable, one of these passes through a further arbitrary point of f, by choosing t properly, and conversely an arbitrary value of determines a single further point of f; the co-ordinates of the points of ƒ are thus rational functions of a parameter, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not (n-3)n but }(n−3)n+1 double points. We may therefore assume that to every point of F corresponds only one point of f, and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n-3 fixed points taken upon f, that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of f, and will not be rational in the coefficients of ƒ unless we can specify a point of f whose coordinates are rational in these. The curve F is intersected by a straight line as+on+c=0 in as many points as the number of unspecified intersections of f with ap+bo1+co=0, that is, 3; or F will be a cubic curve, without double points.

Such a cubic curve has at least one point of inflection Y, and if a 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 YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y=o, and the inflexional tangent at Y is Z=0; the equation of the cubic curve will then be of the form

where (o) denotes the point of inflection.

It thus appears that the co-ordinates of any point of a plane curve, f, of order n with (n-3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, 4, of order m be drawn, passing through the double points of the curve, the values of the argument a at the remaining intersections of with f, have a sum which is unaffected by variation of the coefficients of o, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y2=4x3-g2x-ga, in points (u), (u2), (us), the sum +2+ua is zero, or a period; or the general theorem is a corollary from Abel's theorem proved under 17, Integrals of Algebraic Functions. To prove the result directly for the cubic we remark that the variation of one of the intersections (x,y) of the cubic with the straight line y=mx+n, due to a variation &m, in in m and ", is obtained by differentiation of the equation for the three abscissae, namely the equation

F(x)=4x3-gax-g3−(mx+n)2=0,

that x+x+x=1m2, and hence is {(y1—y)/(x1~x;)}2; so that we have another proof of the addition equation for the function (). From this theorem for the cubic curve many of its geometrical properties, as for example' those of its inflections, the properties of inscribed polygons, of the three kinds of corresponding points, and the theory of residuation, are at once obvious. And similar results hold for the curve of order n with (n-3)n double points.

§ 24. Integrals of Algebraic Functions in Connexion with the Theory of Plane Curves.-The developments which have been explained in connexion with elliptic functions may enable the reader to appreciate the vastly more extensive theory similarly arising for any algebraical irrationality, f(x,y) =0.

The algebraical integrals R(x,y)dx associated with this may as before be divided into those of the first kind, which have no infinities, those of the second kind, possessing only algebraical infinities, and those of the third kind, for which logarithmic infinities enter. Here there is a certain number, p, greater than unity, of linearly independent integrals of the first kind; and this number is unaltered by any birational transformation of the fundamental equation f(x,y)=0; a rational function can be constructed with poles of the first order at +1 arbitrary positions (x,y), satisfying f(x,y) =0, but not with a fewer number unless their positions are chosen properly, a property we found for the case p= 1; and p is the number of linearly independent curves of order n-3 passing through the double points of the curve of order n expressed by f(x,y) =o. Again any integral of the second kind can be expressed as a sum of integrals of this kind, with poles of the first order at arbitrary positions, together with rational functions and integrals of the first kind; and an integral of the second kind can be found with one pole of the first order of arbitrary position, and an integral of the third kind with two logarithmic infinities, also of arbitrary position; the corresponding properties for p=1 are proved above. There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u=/R(x,y)dx be such an integral, it can be shown, in common with all algebraic integrals associated with f(x,y) =0, to have linearly independent additive constants of indeterminateness, the upper limit of the integral cannot therefore, as we have shown, be a single valued function of the value of the integral. The corresponding theorem, if SR.(x,y)dx denote one of the integrals of the first kind, is that the equations determine the rational symmetric functions of the p positions (x1,y1), SR.(x1,y1)dx1+..........+S Ri(x,,¥p)dxp=Ui, it is thus necessary to enter into the theory of functions of several (x,y) as single valued functions of the p variables, u,... Up: independent variables; and the equation f(x,y) =0 is thus not, 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 automorphic functions, which, however, as has been remarked, have, for p>I, an infinite number of essential singularities.

825. Monogenic Functions of Several Independent Variables.— A monogenic function of several independent complex variables u1, . . . u, is to be regarded as given by an aggregate of power series all obtainable by continuation from any one of them in a manner analogous to that before explained in the case of one independent variable. The singular points, defined as the limiting points of the range over which such continuation is possible, may either be poles, or polar points of indetermination, .or essential singularities.

function is expressible as a quotient of converging power scries in A pole is a point (u,. . u) in the neighbourhood of which the -u(0)

vanish at(u),

of these the denominator series D must u), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u),... u)), or if it does, it must be possible to write D-MD,, N=MNo, where M is a converging power series vanishing at ( u”, "....), and No is a converging power series, in (u-u,...up-u), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax+By)/ (ax+by), about x=0, y=0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by (°) (0) making the variables u1,...u, approachtou by a proper path. It is the necessary existence of such polar points of indetermination, which in case p>2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which any one of the variables u1, ... u, is indefinitely great; such points are brought under the preceding definitions by means