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form where B0, B3, etc., are related to b0, b1, b2, ... in the same way as are A0, A1, A2, ... to a0, a1, a2, ... But the expansions (10) and (14) must be identical on account of the uniqueness of the canonical form, so that = A_s, showing that A0, A3, A4, ... are actually absolute invariants. It is not difficult, moreover, to verify this fact by applying the infinitesimal transformations (7) to the expression (II) for A_k. For we find dA_s = 0, which again proves that A_s is an invariant. The invariants A0, A3, A4, ..., A_s, ... are independent functions of the coefficients a0, a1, a2, ... For, taken in this order, each of the invariants involves a coefficient as which does not appear in any of the earlier invariants. Every absolute invariant is a function of A0, A3, A4, ... For, let I(a0, a1, a2, ...) be an absolute invariant. Then we shall have (15) I(a3, a2, ...) = I(Ga, Ga-1, G-2, ...) whenever a0, ap, a2, ... are connected with a9, an, a2, ... by means of the relations (5). In particular, (15) will be verified if we choose the values (9) for c and P, as a result of which choice, the expansion assumes its canonical form. Therefore we shall have I(Ga, a2, a3, ...) = I(Ao, A1, O, A3, A4, ...) showing that I is a function of Ao, A1, A2, ... as stated. We may summarize our last three theorems by saying, that the invariants Ao, A1, A2, ... form a complete system of independent absolute invariants whenever a1 is not equal to zero. We shall ordinarily maintain the hypothesis a1 = 1. If a1 is equal to zero we study, in place of the expansion (i), the Taylor expansion of J(z) for some point z = k where k ≠ o. Of course k may always be chosen in infinitely many ways so as to ensure that the coefficient of z - k in the new expansion will not be equal to zero, provided that w is not a
c and P, as a result of which choice, the expansion assumes its canonical form. Therefore we shall have I(Ga, a2, a3, ...) = I(Ao, A1, O, A3, A4, ...) showing that I is a function of Ao, A1, A2, ... as stated. We may summarize our last three theorems by saying, that the invariants Ao, A1, A2, ... form a complete system of independent absolute invariants whenever a1 is not equal to zero. We shall ordinarily maintain the hypothesis a1 = 1. If a1 is equal to zero we study, in place of the expansion (i), the Taylor expansion of J(z) for some point z = k where k ≠ o. Of course k may always be chosen in infinitely many ways so as to ensure that the coefficient of z - k in the new expansion will not be equal to zero, provided that w is not a constant. It is possible, however, to replace the invariants A0, A3, A4, ... by a new system adapted to the case a1 == a2 == ... == dp-1 = 0, av ≠ o, by an obvious extension of the method which we have used for the case a1 ≠ o. Since we have a j = j • a, is a relative invariant of weight I. Consequently as As is a relative invariant of weight s. According to (I I), the last term of as A, (corresponding to t = s - I) contains as a factor, and no other term in as A-s contains a higher power of a1 in its denominator. Therefore a*-1 As is a relative invariant of weight 2s2, which is an integral rational function of a1, a2, ..., as. Moreover this invariant is a linear function of a1, a2 as. The only coefficients which appear raised to a higher power are a1 and a2. For this reason we shall occasionally refer to the invariants s as the linear invariants of the function f(z). It is now easy to see that these linear invariants form a complete system in a more specific sense than that mentioned
Since we have a j = j • a, is a relative invariant of weight I. Consequently as As is a relative invariant of weight s. According to (I I), the last term of as A, (corresponding to t = s - I) contains as a factor, and no other term in as A-s contains a higher power of a1 in its denominator. Therefore a*-1 As is a relative invariant of weight 2s2, which is an integral rational function of a1, a2, ..., as. Moreover this invariant is a linear function of a1, a2 as. The only coefficients which appear raised to a higher power are a1 and a2. For this reason we shall occasionally refer to the invariants s as the linear invariants of the function f(z). It is now easy to see that these linear invariants form a complete system in a more specific sense than that mentioned above. Every absolute invariant, which is a rational function of the coefficients of f(z), will be a rational function of the linear invariants. The invariant A3 is of special importance. We have so that A3 is equal to the value which assumes for z = o, if we write so that z represents the Schwarzian derivative of w with respect to z. We shall frequently think of an expansion of the function W = f(z) in the neighborhood of a point not at the origin, where f(z) is analytic, W = f(Z) - ao + adZ + 4(Z - z)2 + ... and we shall then write We may then think of ao, a1, a2, ... as well as w, w', W", ... as functions of z and we shall have where a specific determination of the square root being chosen in any particular case. Since A3, A4, A5, ... are the coefficients of z, z, ... in the canonical expansion, it is apparent that A4, A5, ... may be obtained from A3 by repeated differentiation with respect to z. But we may exhibit another and more convenient differentiation process which accomplishes the same purpose. If we make any linear
that z represents the Schwarzian derivative of w with respect to z. We shall frequently think of an expansion of the function W = f(z) in the neighborhood of a point not at the origin, where f(z) is analytic, W = f(Z) - ao + adZ + 4(Z - z)2 + ... and we shall then write We may then think of ao, a1, a2, ... as well as w, w', W", ... as functions of z and we shall have where a specific determination of the square root being chosen in any particular case. Since A3, A4, A5, ... are the coefficients of z, z, ... in the canonical expansion, it is apparent that A4, A5, ... may be obtained from A3 by repeated differentiation with respect to z. But we may exhibit another and more convenient differentiation process which accomplishes the same purpose. If we make any linear transformation of the independent variable, we find and From our knowledge of the fact that A2 is an absolute invariant and from (22), we conclude a familiar formula which may also be verified directly. More generally we make the following remark. If I is a relative invariant of weight p, the transformation (20) will transform I into I where If I is an absolute invariant, we have p = 0, and therefore Thus we may take tv, 0, as a fundamental system of invariants. II. — Integral invariants. All of the invariants obtained so far are functions of the values, w, w', w", which W = f(Z) and its derivatives assume for a given value z of the complex variable Z, the function f(Z) being analytic in the neighborhood of this point. We shall henceforth speak of these invariants as differential invariants. But equation (23) of Art. i enables us to define a new kind of invariant which we shall call an integral invariant of the function. Equation (23) may be written as follows if the square roots are properly determined. Let us select a curve C of finite length in the z plane. This curve may be open or
the transformation (20) will transform I into I where If I is an absolute invariant, we have p = 0, and therefore Thus we may take tv, 0, as a fundamental system of invariants. II. — Integral invariants. All of the invariants obtained so far are functions of the values, w, w', w", which W = f(Z) and its derivatives assume for a given value z of the complex variable Z, the function f(Z) being analytic in the neighborhood of this point. We shall henceforth speak of these invariants as differential invariants. But equation (23) of Art. i enables us to define a new kind of invariant which we shall call an integral invariant of the function. Equation (23) may be written as follows if the square roots are properly determined. Let us select a curve C of finite length in the z plane. This curve may be open or closed but it should be so chosen that f(z), z' is analytic in the neighborhood of each of its points. Let C be the curve in the z plane obtained from C by the transformation (20). Then we shall have Thus, the value of the integral remains unchanged if the independent variable and the path of integration are transformed simultaneously by the same linear transformation. We may express this by saying that o is an integral invariant. More specifically we shall speak of o as a simple integral invariant because the function under the integral sign depends only upon w, w', w", .... The integrand does not itself involve integration. If I is any absolute differential invariant, the integral will be an integral invariant. Conversely, if w is not a linear function of z, any simple integral invariant may be expressed in this form. The truth of the direct statement is apparent. To prove the converse, let J = F(w, w', w", ...) dz be a simple integral invariant. If w is not a linear function of z, the Schwarzian derivative dw, dz will not be equal to zero identically, and we may introduce to as a new independent variable
independent variable and the path of integration are transformed simultaneously by the same linear transformation. We may express this by saying that o is an integral invariant. More specifically we shall speak of o as a simple integral invariant because the function under the integral sign depends only upon w, w', w", .... The integrand does not itself involve integration. If I is any absolute differential invariant, the integral will be an integral invariant. Conversely, if w is not a linear function of z, any simple integral invariant may be expressed in this form. The truth of the direct statement is apparent. To prove the converse, let J = F(w, w', w", ...) dz be a simple integral invariant. If w is not a linear function of z, the Schwarzian derivative dw, dz will not be equal to zero identically, and we may introduce to as a new independent variable in place of z. Then J becomes which is invariant for all possible curves C, we must have dJ = dJ. Since we have also dz = dw, we find which shows that log (z, w) is an absolute differential invariant. There exists an integral invariant even simpler than log (z, w), namely dw, and by an argument analogous to that just completed we can show that any other simple integral invariant may be expressed in the form f dw where I is an absolute differential invariant. In fact this form remains valid even if w is a linear function of z, provided it is not a constant. But the absolute differential invariant of lowest order, excepting w itself, is 0. Therefore log dw is the simple integral invariant of lowest order, excepting only the integral f dw. This latter integral however is related so simply to the values of w at the end points of the path of integration, that we prefer not to think of it as an integral at all. With this understanding we may claim that log is the simplest integral invariant of w = f(z) for linear transformations of the independent variable. III. — The
(z, w), namely dw, and by an argument analogous to that just completed we can show that any other simple integral invariant may be expressed in the form f dw where I is an absolute differential invariant. In fact this form remains valid even if w is a linear function of z, provided it is not a constant. But the absolute differential invariant of lowest order, excepting w itself, is 0. Therefore log dw is the simple integral invariant of lowest order, excepting only the integral f dw. This latter integral however is related so simply to the values of w at the end points of the path of integration, that we prefer not to think of it as an integral at all. With this understanding we may claim that log is the simplest integral invariant of w = f(z) for linear transformations of the independent variable. III. — The intrinsic equation. Let w be given as an analytic function of z. We put the square root being properly specified, and the integration being performed over a path C at all of whose points V dw, is analytic. If we regard o as a function of its upper limit z, and eliminate z, we obtain a relation of the form 0 = F(©), where F is an analytic function of o. We shall speak of this, which expresses the relation between the simplest differential and integral invariants of the function w = f(z), as the intrinsic equation of w = f(z), a terminology which corresponds to that used by Cesaro in his intrinsic geometry. The importance of this notion for our theory appears in the following theorem. All functions, which can be obtained from each other by linear transformation of the independent variable, have the same intrinsic equation. Conversely, if the intrinsic equation is given, there will correspond to it infinitely many functions all of which may be obtained from one of them, w = f(z), by adding a constant and making a linear transformation of z. The truth of the direct theorem is obvious, since 9 and p are
and eliminate z, we obtain a relation of the form 0 = F(©), where F is an analytic function of o. We shall speak of this, which expresses the relation between the simplest differential and integral invariants of the function w = f(z), as the intrinsic equation of w = f(z), a terminology which corresponds to that used by Cesaro in his intrinsic geometry. The importance of this notion for our theory appears in the following theorem. All functions, which can be obtained from each other by linear transformation of the independent variable, have the same intrinsic equation. Conversely, if the intrinsic equation is given, there will correspond to it infinitely many functions all of which may be obtained from one of them, w = f(z), by adding a constant and making a linear transformation of z. The truth of the direct theorem is obvious, since 9 and p are absolute invariants for all linear transformations of z. To prove the converse we observe that so that consequently w is determined as a function of a except for an additive constant. From the definition of o we find: On the other hand we have so that Hence we may replace o by the function of w which results for it from (32), giving dz, dw as a known function of w, say and G (w) will be independent of k. If z is any solution of this equation, the most general solution will be where a, y, z, are arbitrary constants. Thus if w = f(z) is one function which corresponds to the given intrinsic equation, will be the most general function of the same kind. The following method of proving the same theorem is more elegant and leads to some further results. Since (32) furnishes an expression for o as a function of cp by means of a simple quadrature, we are naturally led to think of z also as a function of 9. Now Cayley first proved the formula which shows how to transform the independent variable of a Schwarzian. If we put s = w, x =
o by the function of w which results for it from (32), giving dz, dw as a known function of w, say and G (w) will be independent of k. If z is any solution of this equation, the most general solution will be where a, y, z, are arbitrary constants. Thus if w = f(z) is one function which corresponds to the given intrinsic equation, will be the most general function of the same kind. The following method of proving the same theorem is more elegant and leads to some further results. Since (32) furnishes an expression for o as a function of cp by means of a simple quadrature, we are naturally led to think of z also as a function of 9. Now Cayley first proved the formula which shows how to transform the independent variable of a Schwarzian. If we put s = w, x = z, o'k = 9, we find dw, dz = dV-, dz, [ ? - 1 ]. If w is not a linear function w, z; is not equal to zero, and we conclude dw, d9; = 1. We may write and our theorem now follows immediately from the fact that (32) gives the right member of (37) as a known function of 9 which is independent of k, and that all solutions z of (37) are linear functions of any particular solution. But we may actually indicate the analytic nature of this problem a little more closely. If G = F(s) is the given intrinsic equation, we find from (32) where I is now a known function of ©. Of course the most general solution of (38) is a linear function of w. Consider the linear homogeneous differential equation and let W1, W2 be two of its linearly independent solutions. Then W . W will be a solution of (38). Conversely if w is given as function of ©, as in the present case, we may find W1, W2. Let ©0 be a value of © for which I is analytic and let more especially let the constant k in
(32) gives the right member of (37) as a known function of 9 which is independent of k, and that all solutions z of (37) are linear functions of any particular solution. But we may actually indicate the analytic nature of this problem a little more closely. If G = F(s) is the given intrinsic equation, we find from (32) where I is now a known function of ©. Of course the most general solution of (38) is a linear function of w. Consider the linear homogeneous differential equation and let W1, W2 be two of its linearly independent solutions. Then W . W will be a solution of (38). Conversely if w is given as function of ©, as in the present case, we may find W1, W2. Let ©0 be a value of © for which I is analytic and let more especially let the constant k in equation (32) be equal to zero. Then there exist in the ©-plane a circle of non-vanishing radius around ©0, such that for all points in this circle whence follows if we put and therefore In order to determine z as a function of ©, we must integrate the differential equation Any solution of this equation may be equated to where \(Z_1, Z_2\) are independent solutions of. We may write this equation in the form and integrate it as though the right member were a known function of \(z\), that is, as though (44) were a non-homogeneous linear differential equation whose right member is given. We find in this way that any solution of (44) satisfies the equation where \(W_1\) and \(W_2\) are given by (41), or if we put the arguments into evidence, If \(Z_1\) and \(Z_2\) denote those solutions of (43) which are defined by the same initial conditions (40) which determine \(W_1\) and \(W_2\), \(Z_1\) and \(Z_2\) will satisfy the two integral equations (46) as the most general expression for the independent variable \(z\) which corresponds to the given intrinsic equation. The common kernel of the two integral equations (46) may be written so that (48) \(K(y, z)
Any solution of this equation may be equated to where \(Z_1, Z_2\) are independent solutions of. We may write this equation in the form and integrate it as though the right member were a known function of \(z\), that is, as though (44) were a non-homogeneous linear differential equation whose right member is given. We find in this way that any solution of (44) satisfies the equation where \(W_1\) and \(W_2\) are given by (41), or if we put the arguments into evidence, If \(Z_1\) and \(Z_2\) denote those solutions of (43) which are defined by the same initial conditions (40) which determine \(W_1\) and \(W_2\), \(Z_1\) and \(Z_2\) will satisfy the two integral equations (46) as the most general expression for the independent variable \(z\) which corresponds to the given intrinsic equation. The common kernel of the two integral equations (46) may be written so that (48) \(K(y, z) - K(1|y, 1|)\). Therefore the kernel is skew symmetric. Thus, the general problem of finding the functions which correspond to a given intrinsic equation has been reduced to a problem in linear integral equations of the Volterra type with a skew symmetric kernel. We shall actually determine the intrinsic equations of several important classes of functions later. For our present purposes it suffices to note a few very simple cases. If \(w\) is a linear function of \(z\), but not a constant, \(0\) is identically equal to zero. Thus, the intrinsic equation of linear functions is \(e = 0\). Let us consider the function \(w = az + b\), \(a \neq 0\), we find \(w' = az + b\), \(w" = a\), \(w''' = 0\), and the intrinsic equation is \(0 = 1 = 0\). We shall henceforth speak of functions of the form where \(k, l, m, n, p, y, 0\) are constants, as logarithmic functions. Finally, let \(w = e^{az + b}\), \(a \neq 0\), where \(a\) and \(b\) are constants. We find \(w' = az + b\), \(w" = a^2z + b\), \(w''' = a^3\), whence \(e = b = (e - b) e^{i ?}\) and therefore
of several important classes of functions later. For our present purposes it suffices to note a few very simple cases. If \(w\) is a linear function of \(z\), but not a constant, \(0\) is identically equal to zero. Thus, the intrinsic equation of linear functions is \(e = 0\). Let us consider the function \(w = az + b\), \(a \neq 0\), we find \(w' = az + b\), \(w" = a\), \(w''' = 0\), and the intrinsic equation is \(0 = 1 = 0\). We shall henceforth speak of functions of the form where \(k, l, m, n, p, y, 0\) are constants, as logarithmic functions. Finally, let \(w = e^{az + b}\), \(a \neq 0\), where \(a\) and \(b\) are constants. We find \(w' = az + b\), \(w" = a^2z + b\), \(w''' = a^3\), whence \(e = b = (e - b) e^{i ?}\) and therefore \(e = b = (e - b) e^{i ?}\) and therefore, the intrinsic equation of any exponential function of the form where \(l, a, p, y, 0\) are constants, is \(e = k = 1 = 0\), where \(k\) is a constant, which may be equated to unity if the lower limit of the integral invariant \(o\) be selected accordingly. MINI. — Rational osculants. In order to be able to interpret the invariants already found, and for the purpose of obtaining still others of a more fundamental character, we now introduce a device suggested by differential geometry. In the metric theory of plane curves, for instance, certain osculating curves of a simple character are introduced, such as the tangent, the osculating circle, etc. The properties of these osculants, and their relations to each other constitute the subject matter of differential geometry. But the kind of osculants considered in any part of differential geometry depends essentially upon the group of transformations which, in that branch of geometry, is regarded as fundamental. Thus the osculating circle plays no part in projective differential geometry, since a projective transformation does not, in general, transform a circle into a circle. Our present investigations are in