DIFFERENTIAL use of algebraical calculations as may be possible. Under the former heading we may, with the assumption of a few theorems belonging to the latter, arrange the theory of partial differential equations and Pfaff’s problem, with their geometrical interpretations, as at present developed, and the applications of Lie’s theory of transformationgroups to partial and to ordinary equations; under the latter, the study of linear differential equations in the manner initiated by Riemann, the applications of discontinuous groups, the theory of the singularities of integrals, and the study of potential equations with existence theorems arising therefrom. In order to be clear we shall enter into some detail in regard to partial differential equations of the first order, both those which are linear in any number of variables and those not linear in two independent variables, and also in regard to the functiontheory of linear differential equations of the second order. Space renders impossible anything further than the briefest account of many other matters; in particular, the theories of partial equations of higher than the first order, the function-theory of the singularities of ordinary equations not linear and the applications to differential geometry, are taken account of only in the bibliography. It is believed that on the whole the article will be more useful to the reader than if explanations of method had been further curtailed to include more facts. When we speak of a function without qualification, it is to be understood that in the immediate neighbourhood of a particular set x0, y0, • • of values of the independent variables x, y, • • • of the function, at whatever point of the range of values for x, y, ■ • ■ under consideration x0, y0, ■ ■ may be chosen, the function can be expressed as a series of positive integral powers of the differences x — x0, y-y0, • • •> convergent when these are sufficiently small (see Function, Analytic). Without this condition, which we express by saying that the function is developable about x0, y0„many results provisionally stated in the transformation theories would be unmeaning or incorrect. If, then, we have a set of k functions, /x • -fk of n independent variables x1 • • xn, we say that they are independent when iisk and not every determinant of k rows and columns vanishes of the matrix of k rows and n columns whose r-th row has the constituents dfrjdxl ■ • dfrjdxn; the justification being in the theorem, which we assume, that if the determinant involving, for instance, the first k columns be not zero for x1 = aq0 • • xn~ xn°, and the functions be developable about this point, then from the equations /x = c1 • ■ fk — Ck we can express xl • • Xk by convergent power series in the differences Xk+1 - Xk°+1 • • xn - xn°, and so regard xx • ■ Xk as functions of the remaining variables. This we often express by saying that the equations /x = cl • • fk — Ck can be solved for xx • • x^ The explanation is given as a type of explanation often understood in what follows. We may conveniently begin by stating the theorem : If each of the n functions ^>1 • • <pn of the (n +1) variables0 x1 • ■ xnt be develthe values x-f • • xn°t , the n differential Ordinary opable about of the form dx^dt = (p^tx^ • • xn) are satisfied equations equations by convergent power series xr—xr° +0 {t~ t°)krl + {t- t00f of the Ar2 + • ■ ■ reducing respectively to aq • • xn° when t — t ; first and the only functions satisfying the equations and order. reducing respectively to aq° • • xn° when t = t°, are those determined by continuation0 of these series. If the result of solving these n equations for aq • • xn0 be written in the form aq(aq • • xnt) = aq° • ■ w„(aq • • xnt)=xn°, it is at once ttg 6 evident that the differential equation dfjdt + (f>1dfldx1 + h ousn^rtf31l unc ' ' +ons <Pndfldxn =wQ possesses n integrals, namely, the equation fva ues h " • »> which are developable about the of the hirst ^ (x° ' ' xn°t ) and reduce respectively to aq • • xn order. when t —1°. And in fact it has no other integrals so reducing. Thus this equation also possesses an unique integral reducing when t = t° to an arbitrary function i/^aq • • xn), this integral being i/I&q • • w„). Conversely, the existence of these
EQUATIONS
449
principal integrals aq • • w„ of the partial equation establishes theexistence of the specified solutions of the ordinary equations dxijdt — (pi. The following sketch of the proof of the existence of these principal integrals for the case w = 2 will show the character of more general investigations. Put x for a;-a:0, &c., and consider the equation a(xyt)df jdx + b(xyt)d/jdy = dfldt, wherein the functionsa, b are developable about a: = 0,2 y = 0, t = 0 ; say a(xyt)—a0 + ta1 + ^2/2!+• 2 b(xyt) = b0 + tb1 + t‘ b2j2'.+ - •, so that adjdx + bdjdy = S0 +181 + ^ 1 S2 + • •, where Sr = arddx + brdjdy. In order that /= p0 + tp1 + t'ip2l2l+ ■ •, wherein _p0, p1 are power series in Proof x, y, should satisfy the equation, it is necessary, as we of the find by equating like terms, that p2:=^oih + existence 5jp0, &c., and in general yis+1 = + .qS#*-! + s2o2yq_2 of inte+ • • + 8sPoi where sr=(s!)/(r!)(s-r!). Now compare grals. with the given equation another equation A(xyt)di'jdx + B(xyt)dFldy—dFldt, wherein each coefficient in the expansion of either A or B is real and positive, and not less than the absolute value of the corresponding coefficient in the expansion of a or2 b. In the second equation let us substitute a series F = P0 + fPj + i P2/ 2! -f • •, wherein the coefficients in P0 are real and positive, and each not less than the absolute value of the corresponding coefficientin p0; then putting Ar = Ardjdx + Y)rdjdy, wTe obtain necessary equations of the same form as before, namely, P1 = A0P0, P2 = A0P1 + AjPq, • • • and in general Ps+1 = A0PS + SjAjP*^ + • • +ASP0. These give for every coefficient in Ps.^ an integral aggregate with real positive coefficients of the coefficients in P„ Ps_i, • • -jPo and the coefficients in A and B ; and they are the same aggregates as would be given by the previously obtained equations for the corresponding coefficients in ps+1 in terms of the coefficients in PsPs-i • * Po and the coefficients in a and b. Hence as the coefficients in P0 and also in A, B are real and positive, it follows that the values obtained in succession for the coefficients in Pj, P2, • • are real and positive ; and further, taking account of the fact that the absolute value of a sum of terms is not greater than the sum of the absolute values of the terms, it follows, for each value of s, that every coefficient in ps+1 is, in absolute value, not greater than the corresponding coefficient in Ps+1. Thus if the series for F be convergent, the series for / will also be ; and we are thus reduced to (1), specifying functions A, B with real positive coefficients, each in absolute value not less than the corresponding coefficient in a, & ; (2) proving that the equation Ac?F jdx + Bc?Fjdy = dYjdt possesses an integral P0 + <Pj + £2P2/2! + • • in which the coefficients in P0 are real and positive, and each not less than the absolute value of the corresponding coefficient in p0. If a, b be developable for x, y both in absolute value less than r and for t less in absolute value than B, and for such values a, b be both less in absolute value than the real positive constant M, it is not difficult to verify that we may take A = B = M^1 - —^1 obtain F=r-(r-a:-i/)^l-4^K'^l-^^^
, and
l°g(l-g) J>
and that this solves the problem when x, y, t are sufficiently small for the two p0—x, p0=y. One obvious application of the general theorem is to the proof of the existence of an integral of an ordinary linear differential equation given by the n equations dyjdx=y1, dyljdx=y2, • •, dyn_1jdx=p -ptfn^ - ■ ■ -pny ; but in fact any simultaneous system of ordinary equations is reducible toa system of the form dxildt = <f>i(tx1 • • xn). Suppose we have k homogeneous linear partial equations of the first order in n independent variables, the general equation being a^d/jdxj + • • + aandfldxn = 0, where 0- = 1 • • k, and that slmt ltane we desire to know whether the equations have common U ‘tt ear ~ solutions, and if so, how many. It is to be understood pa ° tJ. . that the equations are linearly independent, which equa a oas implies that k=n and not every determinant of k rows ’ and columns is identically zero in the matrix in which the i-th element of the cr-th row is = l • • n, <r = l • ■ k). Denoting the left side of the cr-th equation by Pff/, it is clear that every common solution of the two equations Pff/=0, Pp/= 0 is also a solution of the equation Pp(P0.;/) - ?„(?,,/) = 0. We immediately find, however, that this is also a linear equation, namely, SHid//dxj=0 where H; = P^a^ - Paapi, and if it be not already contained among the given equations, or be linearly deducible from them, it may be added to them, as not introducing any additional limitation of the possibility of their having common solutions. Proceeding thus with every pair of the original equations, and then with every pair of the possibly augmented system so obtained, and so on continually, we shall arrive at a system of equations, linearly independent of each other and therefore not more than n in number, such that the combination, in the way described, of every pair of them, leads to an equation which is linearly deducible from them. If the number of this so-called complete system is n, the equations give dfjdxl — 0 • • df/dxn — 0, leading to the nugatory result f—a constant. Suppose, then, the number of this system to be r'<n; suppose, further, that from the matrix of the coefficients S. III. - 57
