solvent in positive integral powers about x = a, by (7)=7+(2-a)"7+ ... + sur le calcul des fonctions (Paris, 1806), and Théorie des fonctions (x-2)(d). Such a function, V, we call a variant analytiques (Paris, Prairial, an V); G. Boole, A Treatise on Differ. Then differentiating V in regard to x, and replacing (a) by its ential Equations (London, 1859); and Supplementary Volume value @gin. 1) +...+any, we can arrange dV/dx, and similarly each (London, 1865); Darboux, Leçons sur la théorie générale des of d2V/dx? NV/dxN, where N = n, as a linear function of surfaces, tt. i.-iv, (Paris, 1887–1896): S. Lie, Théorie der transforma the N quantities m, : nay...), ... (1), and lionsgruppen ii. (on Contact Transformations) (Leipzig, 1890). The rethence by elimination obtain a linear differential equation (8) Quantitative or Function Theories for Linear Equations :for V of order N with rational coefficients. This we c. Jordan, Cours d'analyse, t. iii . (Paris, 1896); E. Picard, Traite equation. denote by F=0. Further, each of 71, is expressible d'analyse, tt. ii. and iii, (Paris, 1893, 1896): Fuchs, Various as a linear function of V, 2V/dx, ... N_IV/dxN, with rational co- Memoirs, beginning with that in Crelle's Journal, Bd. Ixvi. P. 121; efficients not involving any of the na coefficients Aij, since otherwise Riemann, Werke, 24 Aufl. (1892); Schlesinger, Handbuch der V would satisfy a linear equation of order less than N, which is Theorie der linearen Differentialgleichungen, Bde. i.-ii. (Leipzig. impossible, as it involves (linearly) the narbitrary coefficients Aij, 1895-1898); Heffter, Einleitung in die Theorie der linearen Differenwhich would not enter into the coefficients of the supposed equation. tialgleichungen mit einer unabhängigen Variablen (Leipzig, 1894); In particular, Yı, . . Y are expressible rationally as linear functions Klein, Vorlesungen über lineare Diflerentialgleichungen der zweilen of w, dw/dx, dN_1w/dxN-1 where w is the particular function Ordnung (Autographed, Göttingen, 1894); and Vorlesungen über (y). Any solution W of the equation F=o is derivable from die hypergeometrische Function (Autographed, Göttingen, 1894); functions $1,.. $n, which are linear functions of y, ... Yn, just Forsyth, Theory of Differential Equations, Linear Equations. as V was derived from 7...na; but it does not follow that these (7) Rationality Group (of Linear Differential Equations) functions $:,: Smare obtained from y ... Yn by a transforma-Picard, Traité d'Analyse, as above, t. iii.; Vessiot, Annales de tion of the linear group A, B, ...; for it may happen that the r'École Normale, série III. t. ix. p. 199 (Memoir); S. Lie, determinant d(31... ;-)/(dy, . • Yn) is zero. In that case Transformationsgruppen, as above, iii. A connected account is -Sn may be called a singular set, and W a singular solution; it given in Schlesinger, as above, Bd. ii., erstes Theil. satisfies an equation of lower than the N-th order. But every solution (8) Function Theories of Non-Linear Ordinary Equations: V, W, ordinary or singular, of the equation F=o, is expressible Painlevé, Leçons sur la théorie analytique des équations différentielles rationally, in terms of w, dw/dx, : . .dN-Iw/dxN_; we shall write, Paris, 1897, Autographed); Forsyth, Theory of Differential Equasimply, V=r(w). Consider now the rational irreducible equation lions, Part ii., Ordinary Equations not Linear (two volumes, ii. and iii.) of lowest order, not necessarily a linear equation, which is satisfied (Cambridge, 1900); Königsberger, Lehrbuch der Theorie der Differenby w; as y, ... Yn are particular functions, it may quite well tialgleichungen (Leipzig, 1889); Painlevé, Leçons sur l'intégration be of order less than N; we call it the resolvent equation, suppose it des équations differentielles de la mécanique ei applications Paris, of order P, and denote it by y(). Upon it the whole theory turns. 1895). In the first place, as y(0) = 0 is satisfied by the solution w of F =0, all @ Formal Theories of Partial Equations of the Second and Higher the solutions of y(s) are solutions F-o, and are therefore rationally Orders:-E. Goursat, Leçons sur l'intégration des équations aux expressible by w; any one may then be denoted by r(w). If this dérivées partielles du second ordre, tt. i. and ii. (Paris, 1896, 1898); solution of F=o be not singular, it corresponds to a transformation Forsyth, Treatise on Differential Equations (London, 1889); and A of the linear group (A, B, ...), effected upon y, ... Yo The Phil. Trans, Roy, Soc. (A.), vol. cxci. (1898), pp. 1-86. coefficients Aij of this transformation follow from the expressions () See also the six extensive articles in the second volume of before mentioned for ni...na in terms of V, dv/dx, d-v/dx, ... by the German Encyclopaedia of Mathematics. (H. F. BA.) substituting V=r(w); thus they depend on the p arbitrary para- DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, charmeters which enter into the general expression for the integral of acterized by a shell formed of sand granules cemented together; the equation y(u) =0. Without going into further details, it is then these are swallowed by the animal, and during the process of that any solution is expressible rationally, with p parameters, in bud-fission they pass to the surface of the daughter-bud and terms of the solution w, enables us to define a linear homogeneous are cemented there. Centropyxis (Steia) and Lecqueureuxia group of transformations of ... Yn depending on p parameters; (Schlumberg) differ only in minor points. and every operation of this continuous) group corresponds to a DIFFRACTION OF LIGHT.-1. When light proceeding from rational transformation of the solution of the resolvent equation. This is the group called the rationality group, or the group of trans- a small source falls upon an opaque object, a shadow is cast upon formations of the original homogeneous linear differential equation. a screen situated behind the obstacle, and this shadow is found to The group must not be confounded with a subgroup of itself, be bordered by alternations of brightness and darkness, known the monodromy group of the equation, often called simply the group as “ diffraction bands.” The phenomena thus presented were ing on arbitrary variable parameters, arising for one particular described by Grimaldi and by Newton. Subsequently T. Young fundamental set of solutions of the linear equation (see Groups, showed that in their formation interference plays an important The importance of the rationality group consists in three proposi- | Later investigations by Fraunhofer, Airy and others have part, but the complete explanation was reserved for A. J. Fresnel. tions. (1) Any rational function of yi, . . y which is unaltered in value by the transformations of the group can be written greatly widened the field, and under the head of " diffraction" in rational form. (2) If any rational function be changed are now usually treated all the effects dependent upon the in form, becoming a rational function of y. ... Ynu a limitation of a beam of light, as well as those which arise from la regard leave its value unaltered. (3) Any homogeneous linear transformation of the group applied to its new form will irregularities of any kind at surfaces through which it is transtransformation Icaving unaltered the value of every mitted, or at which it is reflected. 2. Shadows. In the infancy of the undulatory theory the belongs to the group. It follows from these that any objection most frequently urged against it was the difficulty of properties (1) (2) is identical with the group in question. It is clear explaining the very existence of shadows. Thanks to Fresnet that with these properties the group must be of the greatest import and his followers, this department of optics is now precisely the ance in attempting to discover what functions of x must be regarded as one in which the theory has gained its greatest triumphs. The rational in order that the values of y . . . Y may be expressed. principle employed in these investigations is due to C. Huygens, And this is the problem of solving the equation from another point and may be thus formulated. If round the origin of waves an LITERATURE.—(a) Formal or Transformation Theories for Equations ideal closed surface be drawn, the whole action of the waves in the of the First Order :-E. Coursat, Leçons sur l'intégration des équa- region beyond may be regarded as due to the motion continually lions aux dérivées partielles du premier ordre (Paris, 1891); E. v. propagated across the various elements of this surface. The wave Weber, Vorlesungen über das Pfaff'sche Problem und die Theorie der motion due to any element of the surface is called a secondary 6. Lie und G. Scheffers, Geometrie der Berührungstransformationen, wave, and in estimating the total effect regard must be paid to the Bd. i. (Leipzig, 1896): Forsyth, Theory of Differential Equations, phases as well as the amplitudes of the components. It is usually Part i., Exact Equations and Pfaff's Problem (Cambridge, 1890); convenient to choose as the surface of resolution a wave-front, i.e. S. Lie, "Allgemeine Untersuchungen über Differentialgleichungen, die a surface at which the primary vibrations are in one phase. Any Annal. xxv. (1885), pp. 71-151; 5. Lie und G. Scheffers, Vorlesungen obscurity that may hang over Huygens's principle is due mainly to über Differentialgleichungen mit bekannten infinitesimalen Transforma- the indefiniteness of thought and expression which we must be lionen (Leipzig, 1891). A very full bibliograpliy is given in the book content to put up with if we wish to avoid pledging ourselves as of E. v. Weber referred to; those here named are perhaps sufficiently to the character of the vibrations. In the application to sound, representative of modern works. Of classical works may be named: where we know what we are dealing with, the matter is simple géométrie (par M. Liouville, Paris, 1850); . L. Lagrange, Legons often stand in the way of the calculations we might wish to make. Warko Supplementband: G Monge, Application de l'analyse a lacnough in principle, although mathematical difficulties would The fundamental theorem to the ratlos. ality group. *S. sink{al-e)x da Fig. 1. The ideal surface of resolution may be there regarded as a flexible By this expression, in conjunction with the qụarter-period acceleralamina; and we know that, if by forces locally applied every tion of phase, the law of the secondary wave is determined. element of the lamina be made to move normally to itself exactly to be expected from considerations respecting energy, but the as the air at that place does, the external aerial motion is fully occurrence of the factor A4, and the acceleration of phase, have determined. By the principle of superposition the whole effect sometimes been regarded as mysterious. It may be well therefore may be found by integration of the partial effects due to each to remember that precisely these laws apply to a secondary wave element of the surface, the other elements remaining at rest. of sound, which can be investigated upon the strictest mechanical principles. We will now consider in detail the important case in which uniform The recomposition of the secondary waves may also be treated plane waves are resolved at a surface coincident with a wave-front analytically. If the primary wave at-o be cos kal, the effect of the (OQ). We imagine a wave-front divided secondary wave proceeding from the element dS at Q is cos k(at-p+IN) = sin k(at-p). -kf sin k(at –p)dp- [ -cos (al-o)]:. In order to obtain the effect of the primary wave, as retarded by Now the effect upon P of each element of the traversing the distance 5, vizcos k(ai -r), it is necessary to suppose depends also upon the distance from P, and possibly upon the portant to notice that without some further understanding the inclination of the secondary ray to the direction of vibration and integral is really ambiguous. According to the assumed law of to the wave-front. the secondary wave, the result must actually depend upon the The latter question can only be treated in connexion with the precise radius of the outer boundary of the region of integration, stances the result is independent of the precise answer that may be very special and exceptional. We may usually suppose that a large siven. An that it is necessary to assume is that the effects of the number of the outer rings are incomplete, so that the integrated term successive zones gradually diminish, whether from the increasing at the upper limit may properly be taken to vanish. It a forma! obliquity of the secondary ray or because (on account of the limita proof be desired, it may be obtained by introducing into the integral tion of the region of integration) the zones become at last more and a factor such as hp, in which h is ultimately made to diminish more incomplete. The component vibrations at P due to the without limit. successive zones are thus nearly equal in amplitude and opposite in When the primary wave is plane, the area of the first Fresnel phase (the phase of each corresponding to that of the inhnitesimal zone is t, and, since the secondary waves vary as the intensity circle midway between the boundaries), and the series which we have is independent of , as of course it should be. if, however, the to sum is one in which the terms are alternately opposite in sign primary wave be spherical, and of radius a at the wave-front of and, while at first nearly constant in numerical magnitude, gradually resolution, then we know that at a distance ? further on the diminish to zero. In such a series each term may be regarded as very amplitude of the primary wave will be diminished in the ratio and thus the sum of the whole series is represented by hall the first area of the first Fresnel zone. For, if : be its radius, we have term, which stands over uncompensated. The question is thus {(r+$1)8-*) tv lai-**] =rta, reduced to that of finding the effect of the first zone, or central so that circle, of which the area is rar. * = lar/(a+r) nearly. We have seen that the problem before us is independent of the Since the distance to be travelled by the secondary waves is still law of the secondary wave as regards obliquity; but the result of, we see how the effect of the first zone, and therefore of the whole the integration necessarily involves the law of the intensity and series is proportional to a/la+r). In like manner may be treated phase of a secondary wave as a function of r, the distance from the other cases, such as that of a primary wave-front of unequal principal origin. And we may in fact, as was done by A. Smith (Camb. Math. curvatures. Journ., 1843, 3, p. 46), determine the law of the secondary wave, by The general explanation of the formation of shadows may also comparing the result of the integration with that obtained by sup be conveniently based upon Fresnel's zones. If the point under posing the primary wave to pass on to P without resolution. consideration be so far away from the geometrical shadow that a Now as to the phase of the secondary wave, it might appear large number of the earlier zones are complete, then the illuminanatural to suppose that it starts from any point with the phase tion, determined sensibly by the first zone, is the same as if there of the primary wave, so that on arrival at P, it is retarded by the were no obstruction at all. If, on the other hand, the point be well amount corresponding to QP. But a little consideration will prove immersed in the geometrical shadow, the earlier zones are altogether that in that case the series of secondary waves could not reconstitute missing, and, instead of a series of terms beginning with finite the primary wave. For the aggregate effect of the secondary waves numerical magnitude and gradually diminishing to zero, we have 1 the hall of that of the first Fresnel zone, and it is the central now to deal with cne of which the terms diminish to zero al both element only of that zone for which the distance to be travelled is ends. The sum of such a series is very approximately zero, each term equal to !. Let us conceive the zone in question to be divided being neutralized by the halves of its immediate neighbours, which into infinitesimal rings of equal area. The effects due to each of are of the opposite sign. The question of light or darkness then these rings are equal in amplitude and of phase ranging uniformly depends upon whether the series begins or ends abruptly. With few over hall a complete period. The phase of the resultant is midway exceptions, abruptness can occur only in the presence of the first between those of the extreme elements, that is to say, a quarter of term, viz. when the secondary wave of least retardation is unoba period behind that due to the element at the centre of the circle. structed, or when a ray passes through the point under consideration, It is accordingly necessary to suppose that the secondary waves According to the undulatory theory the light cannot be regarded start with a phase one-quarter of a period in advance of that of the strictly as travelling along a ray; but the existence of an unobstructed primary wave at the surface of resolution. ray implies that the system of Fresnel's zones can be commenced, Further, it is evident that account must be taken of the variation and, if a large number of these zones are fully developed and do not of phase in estimating the magnitude of the effect at P of the first terminate abruptly, the illuinination is unaffected by the neighbourzone. The middle element alone contributes without deduction; hood of obstacles. Intermediate cases in which a few zones only are the effect of every other must be found by introduction of a resolv- formed belong especially to the province of diffraction. ing factor, equal to cos , if 0 represent the difference of phase An interesting exception to the general rule that full brightness bet ween this element and the resultant. Accordingly, the amplitude requires the existence of the first zone occurs when the obstacle of the resultant will be less than if all its components had the same assumes the form of a small circular disk parallel to the plane of phase, in the ratio the incident waves. In the earlier half of the 18th century R. Delisle + fr found that the centre of the circular shadow was occupied by a cos Odo : , bright point of light, but the observation passed into oblivion until s. D. Poisson brought forward as an objection to Fresnel's or 2: -. Now a area 12-31; so that, in order to reconcile the bright as if no obstacle were intervening. If we conceive the primary theory that it required at the centre of a circular shadow a point as amplitude of the primary wave (taken as unity) with the hall effect wave to be broken up at the plane of the disk, a system of Fresreel's of the first zone, the amplitude, at distances of the secondary wave zones can be constructed which begin from the circumference: emitted from the element of area dS must be taken to be and the first zone external to the disk plays the part ordinanly IS/N (1). Laken by the centre of the entire system. The whole ollect is the of the existence It is seen at once systems, total ditferential wherein perpi.i. are power series in x, y, should satisfy the equa- have in common n- solutions, say 6+1, ... w which reduce tion, it is necessary, as we find by equating like terms, that respectively to Xr+1, ... Xn when in them for xi, x, are respecP1 = 8.por pz = 8.pi+81po, &c. tively put Xi',...X°; so that also the equations have in common a Proof and in general solution reducing when x1 = x;", ... X.= xto an arbitrary function ,Pori = 8.p. + 581 po-17 sgözpo-st... +8.Po, (Xr+1, ... *n) which is developable about *,*+1, . . . Xno, namely, of intewhere S = (s!7(!) (s-r)! this common solution is wrth, ... wn). Now compare with the given cquation another equation that this result is a generalization of the theorem for r=1, and its grals. A(xyl)dF/dx +B(x31)dF/dy=dF/dt, proof is conveniently given by induction from that case. It can be wherein each coefficient in the expansion of cither A or B is real and verified without difficulty (1) that is from the r cquations of the positive, and not less than the absolute value of the corresponding complete system we form → independent linear aggregates, with cuefficient in the expansion of a or b. In the second cquation let us coefficients not necessarily constants, the new system is also a comsubstitute a series plete system; (2) that is in place of the independent variables F=P.+{P.+FP/2!+ ... 21, ... *n we introduce any other variables which are independent wherein the cocfficients in P. are real and positive, and cach not less functions of the former, the new cquations also form a complete than the absolute value of the corresponding cocfficient in po; then system. It is convenient, then, from the complete system of putting 4-=A,d/dx+B.d/dy we obtain necessary cquations of the equations to form 7 new equations by solving separately for dfidx,.., same form as before, namely, P, -A.P., P, AP, +4.Po, ... df/dxr; suppose the general equation of the new systeni to be lof=djidxg+Coireidf/dxit... condf/dx, = 0(0 = 1, ...). and in general Pot1=4.P.+514, P-+...+4.P. These give for Then it is casily obvious that the cquation QQ-0-0-f=o conevery.coefficient in Ps+! an integral aggregate with real positive tains only the differential coefficients of f in regard to xr+l..... as coefficients of the coefficients in P., P-1, ... , P. and the coefficients it is at most a linear function of Qf, ... Q./, it must be identically in A and B; and they are the same aggregates as would be given by zero. So reduced the system is called a Tucobian system. Of this the previously obtained equations for the corresponding coefficients system Of=0 has n-i principal solutions reducing rein pati in terms of the coefficients in po po-u; ... De and the co-spectively to xa, ... In when Jacobian efficients in a and b. Hence as the coefficients in P, and also in A, B are real and positive, it follows that the values obtained in succession and its form shows that of these the first r-1 are exactly x2 ..., taking account of the fact that the absolute value of a sum of terms independent variables in all the r equations. Since the first equation is not greater than the sum of the absolute values of the terms, it is satisfied by n-1 of the new independent variables, it will contain follows, for cach valuc of $, that every cocfficient in Doris, in absolute no differential coefficients in regard to them, and will reduce therefore value, not greater than the corresponding coetñcient in polThus simply to df/dx1 =0, expressing that any common solution of the if the series for F be convergent, the scrics for f will also be; and we equations is a function only of the n-i remaining variables. Thereby are thus reduced to (I), specifying functions A, B with real positive the investigation of the common solutions is reduced to the same coefficients, each in absolute value not less than the corresponding problem for r-1 equations in n-i variables. Proceeding thus, we coefficient in a, 6; (2) proving that the equation reach at length one equation in n-sti variables, from which, by in P. are real and positive, and each not less than the absolute valuc With the coefficients Cay of the cquations Def=o in transposed possesses an integral P. +{P.+FP2/21+ ... in which the coefficients retracing the analysis, the proposition stated is seen to follow. The analogy with the case of one equation is, however, still closer, both in absolute value less than rand for tless in absolute value than in-uy equations, 22, =cıqda;+: "+rds, equivalent to System ol R, and for such values a, o be both less in absolute value than the ther(n-) equations dx,/dxo = com. That consistent real positive constant M, it is not difficult to verify that we may with them we may be able to regard tralo .... X as take A=B=M (1-72) "(-x)" and obtain functions of X, ... X, these being regarded as independent equations. variables, it is clearly necessary that when we differentiate Caj in regard to Xp on this hypothesis the result should be the same as when R, we differentiate Cpi in regard to so on this hypothesis. The differand that this solves the problem when x, y, I are sufficiently small ential coefficient of a function of x. ... X on this hypothesis, in for the two cases po=x, Dory, One obvious application of the regard to xp, is, however, general theorem is to the proof of the existence of an integral of df/dxpt Cportidf/dX+1 + ... topndfdxn, an ordinary linear differential cquation given by the n equations namely, is def. Thus the consistence of the n-7 total cquations dy/dx=yı, dy/dx= y..,, ruquires the conditions pro; - Qarp=0, which arc, however, dy-i/dx = p-piyr-1...-Py; but in fact any simultaneous system of ordinary equations is re- easily verified that it wr+1, verified in virtue of 0.(Qa) - Q.10.B=0. And it can in fact be .. wn be the principal solutions of the ducible to a system of the form Jacobian system, Qof=0, reducing respectively to Xr+1, ... when dx./dt = 0.(iti, ... X.). Suppose we have k homogencous lincar partial cquations of the be solved for $2.Xm to givex;=;(*1,*.***..*."), these first order in n independent variables, the general cquation being values solve the total equations and reduce respectively to x.* *!....X® SimultaneQuidf/dx.+...+ao,df|dx, =0, where a = 1, ... k, and that x=x". And the total equations have no we desire to know whether the equations have common ous lgear solutions, and if so, how many. It is to be understood of these solutions of the total equations can be deduced a priori other solutions with these initial values. Conversely, the existence that the equations are linearly independent, which implies and the theory of the Jacobian system based upon them. The is identically zero in the matrix in which the i-th clement of the oth theory of such total equations, in general, finds its natural place row is oo.(i=1, ... , =1,... k). Denoting the left side of the A practical method of reducing the solution of the r equations o-th equation by Pof, it is clear that every common solution of the two equations Pof=0, P=0 is also a solution of the equation of a Jacobian system to that of a single equation in N-:+I variables may be explained in connexion with a geometrical interP.(P.1)-P.(P) = o. We immediately find, however, that this is pretation which will perhaps be clearer in a particular also a linear equation, namely, 211,dfidei = where Hi = Ppden-Podpis case, say n=3, y=2. There is then only one total cal later and if it be not already contained among the given equations, or be equation, say ds=ads+bdy; if we do not take account ducing any additional limitation of the possibility of their having dady+bda/ds = db/dx tadb]d2, this equation may be recommon solutions. Proceeding thus with every pair of the original garded as defining through an arbitrary point (te, yo. 2.) of three equations, and then with every pair of the possibly augmented dimensioned space (about which a, b are developable) a plane, namely, system so obtained, and so on continually, we shall arrive at a 2-3, -a.(x-x.)+b.(y-y), and therefore, through this arbitrary system of equations, linearly independent of cach other and therefore point o directions, namely, all those in the plane. If now there be not more than n in number, such that the combination, in the way a surface 2=\(x, y), satisfying ds=ads+bdy and passing through described, of every pair of them, leads to an equation which is lincarly deducible from them. If the number of this so-called (co, Yo, 5o), this plane will touch the surface, and the operations of complete system is n, the equations give df/dx, =o... didx=0, passing along the surface from (x., Y; 2.) to (+.+dxo, Yo, 2+dz.) leading to the nugatory result f=a constant. Suppose, then, the number of this system to be r. <n; suppose, further, that from the and then to (*.+dx., %:+dya, 2+d'2.), ought to lead to the same matrix of the coefficients a determinant of r rows and value of d's, as do the operations of passing along the surface from Complete columns not vanishing identically is that formed by the (xo, Y., 2.) to (, +dye, %. +8.), and then to (x+xa, y.+dy, 2, +812.). d'a=0.dx. +6(x+dxw, Yo, I ta.dx.)dy, - adxo+b.dy.+dx.dy (lot.). tions, and therefore the originally given sct of k equations, and so at once reach the condition of integrability. If now we put F when xy=xio partial equations. Geometri and solution. systems of linear partial equations Mayer's integra tiloa. Pallian taneous Plaffian #=x+4y=yo+mt, and regard m as constant, we shall in fact be so that the equation considering the section of the surface by a fixed plane y-y. =(x-x.); dy-udh-...-udyruotidiotra...-undln = 0 dr'da+bm, where a, b are expressed as functions of mand, with is identically true in regard to w, .... Un test... !a: equating to - kept constant, finding the solution which reduces to 2. for t=0, zero the coefficients of the differentials of these variables, we thus and in the result again replace m by (y-y.)/(x-xo), we shall have the obtain m-s relations of the form surface in question. In the general case the cquations dy/d1,-1,duld...-udyo/dt,-1;=0; dx; - Cı,dxi +...(dxr similarly determine through an arbitrary point'*7', ... *° of m +1 relations connecting the 2m +1 variables in virtue of which these m-s relations, with the previous sti relations, constitute a set method of a planar manifold of, dimensions in space of n dimensions, the Plaffian equation is satisfied independently of the form of the and when the conditions of integrability are satisfied, functions h... There is clearly such a set for cach of the every direction in this manifold through this point is values s = 0,5-1, ..., s=m-1, s=m. And for any value of s there tangent to the manifold of r dimensions, expressed by Wr+1=*:may exist relations additional to the specified m+1 relations, pro w.**.*, which satisfies the equations and passes through this vided they do not involve any relation connecting 1,4, ...hmonly, point. If we put 11-4° = 1, xx° = mg, . . *-Xo = mel; and and are consistent with the m-s relations connectingu. ... Blom. It regard ma.... m, as fixed, the (n-) total equations take the form is now evident that, essentially, the integration of a Pfaffian equation ds, di-cut maca, +...+mers, and their integration is equivalent to that of the single partial cquation andxg+... tandxn=0, wherein aj, ... , are functions of x, ... Xn, is effected by the ef/de+ z(cu+mast... +mendf/dx;=0 processes necessary to bring it to its reduced form, involving only independent variables. And it is easy to see that if we suppose this in the n-r+! variables 1, 3+1, . . . * Determining the solutions reduction to be carried out in all possible ways, there is no need to Boyle....la which reduce to respectively Xr+1,...x, when t=0, and sub-distinguish the classes of integrals corresponding to the various Kituting. I = x1-x1°, ma = (xp-x3")/(x1-31"). ...m= (x,-*°)/(x,-*"), values of s; for it can be verificd without difficulty that by putting we obtain the solutions of the original system of partial equa-1!=-11-...de, = 1, ...l'. = 4 u's--, ..., u'.--la tions previously denoted by way.... Ww. It is to be remarked, *1 lb...l'-lu'. = 4!!....unumi the reduced cquation however, that the presence of the fixed parameters m. ... m, in becomes changed to dt'u',di'r-. ..-u'ndt' =0, and the general the single integration may frequently render it more difficult than it relations changed to they were assigned numerical quantities. t=(tel....!'..) - tiltot....!')-...-Pt.(l'... ...l'.), , We have above considered the integration of an equation say, together with u' -do/de", ...,' mdodi', which contain only ds=ads+bdy one relation connecting the variables t', ', ...1's only: on the hypothesis that the condition This method for a single Pfaffian equation can, strictly speaking, daldy+bda/ds -dbdstadb/ds. be generalized to a simultancous system of (?-?) l'affian equations It is natural to inquire what relations among x, y, s, if any, dx=dxt...tc,,dx, only in the case already treated, Simulare implied by, or are consistent with a differential relation when this systern is satisfied by regarding Xpl...., as adx +bdy+cdx0, when a, b, c are unrestricted functions suitable functions of the independent variables x.. ...; of x, y, s. This problem leads to the consideration of the in that case the integral manifolds are of, dimensions equations so-called Pfaffian Expression adx+bdy+cdz. It can be shown (1) if When these are non-existent, there may be integral manieach of the quantities db/ds-dc.dy, dc/dx-dud:, daldy-dbde, which folds of higher dimensions; for if we shall denote respectively by us, a, 113, be identically zero, the do= 0,2xy +...+ dx,+Dr+(c10+idxit...touldxx) +€4( )+... expression is the differential of a function of x, 7. 5, equal to di say: be identically zero, then potco+10+1+...+co,non = 0, or o satisfies (2) that if the quantity qua+bua +cus is identically zero, the ex. the partial differential cquations previously associated with the pression is of the form udt, i.e. it can be made a perfect differential total equations; when these are not a complete system, but in pression is of the form ditudes. Consider the matrix of four ----- independent integrals, the total cquations are satisfied over rows and three columns, in which the elements of the first row are a manifold of rtu dimensions (see E. v. Weber, Math. Annal. lv. 4.6., and the elements of the (+1)-th row, for on 1, 2, 3. are the (1901), p. 386), seen that the cases (1), (2), (3) almve correspond respectively to the character, which naturally arise in connexion with the theory of It seems desirable to add here certain results, largely of algebraic ass when (1) every determinant of this matrix of two rous and contact transformations. For any two functions of the 2n columns is zero, (2) every determinant of three rows and columns independent variables x1, ... X, P... Po we denote by (ou) is zero, (3) when no condition is assumed. This result can be general daily dido ized as follows: if a, .... be any functions of ... ... ... the so- the sum of the n terms such as For two mations called Pfaffian expression (dx+...+andx. can be reduced to one functions of the (2n +1) independent variables 2.41, ...fo, pl.... pa dp.dk, dp.ds. or other of the two forms we denote by lov) the sum of the x terms such as Hidhit...tudis, ditu,dlit...tu-dla-, wherein !, *. ..., .... are independent functions of $1,...X.. and k do dy do is such that in these two cases respectively ak or 2-1 is the rank of a certain matrix of n+1 rows and n columns, that is, the greatest It can at once be verified that for any three functions lovl]+($14/11 manier the matrix is that whore first row is constituted by the +1448611 462) + Sol , which when 5,9,4 do not contains quantity ds./dxrdagider. The proof of such a reduced form can becomes the identity (lov))+(())+(()) =0. Then, if X:....X. be obtained from the two results: (1) If I be any given function P, : P, be su h functions of x1, . *.. ... Pathat PdX; of the am independent variables ...., .... In the expression +...+PdX, is identically equal to pidx! + ... +p_dx.. it can be d+di+...+ndle can be put into the form a'd' +...+undt' shown by clementary algebra, alter equating coefficients of inde(2) If the quantities W..., , b... Lan be connected by a relation, penlent (lifferentials, (1) that the functions X.... P. are independthe expression mıdli +...+ dia can be put into the form di' + widt' ent functions of the 2n variabiles xs, ... pa so that the equations +...+x'mzdim; and if the relation connecting ..... Mm.blox.=X.. P.-P. can be solved for xu....X.. Pl.... pand represent be homogeneous in u. ... #m. then i' can be taken to bre zero. These therefore a transformation, which we call a homogrneous contact tvorosults are decurtions from the theory of contact transformations transformation; (2) that the X.... X. are homensen ous functions of Ive below), and their demonstration requires, beside elementary ..pa of zero dimensions the P.... Pare homenerus functions alebraical considerations, only the theory of complete systems of lofp... : Prof dimension one, and the An(n-1) hlutions (X.X,) - 0 lancar humogeneous partul diferential equations of the first order. are verified. So als) are the morelations (IX.) -1, (P.X,)-0, When the existence of the reluced form of the Maffian expression (?".P,) no. Conversely, if X1.... X, loc independent functions, cach containing only independent quantitie, is thus core assured, the humnymous of zer dimensiun in P.... Nirying the JH(N-1) identification of the numlar k with that defined by the specified relation. (X,X,) -0, then l'.... P. can bx uniquely determined, by matrix muy, with me dathculty, lve made a foulerturi. solving linear algebraic muations, such that 1.4X;+...+P.dx. In all cases of a single l'affian cquation we are thu led to consider - pidxi+...+p.dr. If now we put nti fur . put : for X... what is implıcd by a relation ds-ud-...-unda-9, in which ? lur X... Q. for-P'./P... for 11, ... , put g. for-paper and o independent variables. This is to biti fiext in virtue of las cuenta . .......: 91....4*, ur obtain the fulluwing results: 11 2X, X., .... one or yrveral relation connecting the variable; there fue functions of :, *.. ..... .... such that the expression must involve relatie conne ting I, 1, ... only, and dl-PX; -...-PdX, is identially equal to olds-pdr...:-peds.), in one of these at kalmus drtually enter. Wecan and not z't, then (1) the function 2, X, ... X., P. then suppose that in une actual tem of relations in virtue of which are in lependent funtion : 1......... Pa, so that the the Maihan cuation is satisfied, all the relations connecting 1,1 ... muitions :'-Zx', -X.pi-Pican loc welur ?,.....po L only are given by and determine a transformation which we call a mun bomogeneous ilm...de), bohlen...hm), ..., -dul...lm); contact transformation; (2) the 2, X.. ... X, verily the nintil Contact transfor Plaflies equation dP Partial identities (ZX.] =0, (X,X,)=0. And the further identities relations G=0, H=o connecting x, y, 2, 8, 9 and independent of (PX)=, (Px;l=0, (P.2)=GP, [P;P,)=0, F= 0, so that the three relations together may involve 120)-2--,(X:0) --*P.) da= pdx+qdy. are also verified. Conversely, if Z, X., 4. X, be independent func :=$(x, y), p=dy/dx, q=d4/dy; but it may also, as another Such a set of three relations may, for example, be of the form zero, and P. P. can be uniquely determined, by solution of case , involve two relations s=46), x=4i(y) connecting x, y, 2, algebraic equations, such that the third relation being dz-PdX,-...-P.dX, = o(dz-pidx- ...- podxn). V'(y)= ÞV1(y)+9, Finally, there is a particular case of great importance arising when the connectivity consisting in that case, geometrically, of a curve = 1, which gives the results: (1) If U, X,, . : X., P,, ... P, be in space taken with o 1 of its tangent plancs;. or, finally, a 2n+i functions of the 2n independent variables X., ...*n Dll connectivity is constituted by a fixed point and all the planes ... Pu, satisfying the identity dU+P.dxı+...+P,dX,= pidxıt... +podxa, passing through that point. This generalized view of the meanthen the 2n functions P, ... P.,x, ...x, are independent, ing of a solution of Fro is of advantage, moreover, in view of anomalies otherwise arising from special forms of the equation (X.X;)=0,(X.U)=8X.(P.X:)=1,(PxXx)=0,(P:P,)0,=(P,Y) +P;=P itself. For instance, we may include the case, some- Meaning where : denotes the operator pidlapit... +Pad/dpai (2) If times arising when the equation to be solved is obtained of a soluX1, ... X., be independent functions of 31, ; n, bu. Pri by transformation from another equation, in which F tion of the such that (X;X;)=0, then U can be found by a quadrature, such that does not contain either porq. Then the equation has equation, and when X:, . ... X., U satisfy these {n(n+!) conditions, then 0 ? solutions, each consisting of an arbitrary point of the surface P, ... P, can be found, by solution of linear algebraic equations, to F=o and all the co? planes passing through this point; it also render true the identity du+P,dX.+...+P,dX, -pidx +...+pudx; has co 2 solutions, each consisting of a curve drawn on the surface (3) Functions X, ... Xn, P,... P, can be found to satisfy F=o and all the tangent planes of this curve, the whole consisting this differential identity when U is an arbitrary given function of of co2 elements; finally, it has also an isolated (or singular) to see what integrations, it is only necessary to verify the statement solution consisting of the points of the surface, each associated that if U be an arbitrary given function of x1, ... Xn D ... Pay with the tangent plane of the surface thereat, also co? clements in and, for r<n, X1.... Xr be independent functions of these vari- all. Or again, a linear equation F=Pp+Q9-R=0, wherein the 1+1 homogeneous lincar partial differential equations of the P, Q, R are functions of x, y, z only, has co2 solutions, each first order (Un +8f =0, (Xpf) =0, form a complete system. It will consisting of one of the curves defined by be seen that the assumptions above made for the reduction of dx/P=dy/Q=dz/R Pfaffian expressions follow from the results here enunciated for taken with all the tangent planes of this curve; and the same contact transformations. We pass on now to consider the solution of any partial equation has co ? solutions, each consisting of the points of a differential equation of the first order; we attempt to explain this surface. And for the case of n variables there is similarly surface containing colof these curves and the tangent plancs of certain ideas relatively to a single equation with any the possibility of nti kinds of solution of an equation tial equa. ordinary equation of the first order with one inde F(x1, ... *n, 2, P1, ... Pa)=0; these can, however, by a tion of the pendent variable) by speaking of a single equation with simple contact transformation be reduced to one kind, in which two independent variables x, y, and one dependent there is only one relation :'=*(x' , . . . x'n) connecting the variable 3. It will be seen that we are naturally led to just as in the case of the solution new variables x'ı, . . . x'x, s' (see under Pfaffian Expressions); consider systems of such simultancous equations, which we consider below. The central discovery of the transformation z=(y), x=4i(y), V(y)=pV':(y) +9 theory of the solution of an equation F(x, y, 2, dz/dx, dz/dy)=0 of the equation Pp+Q9=R the transformation :'=2-px, is that its solution can always be reduced to the solution of x'=P, p= -2, y'=y, d' =q gives the solution partial equations which are linear. For this, however, we must d=46)+x'ti(y), p = dz'/dx', d'=ds'/dy' regard dz/dx, ds/dy, during the process of integration, not as the of the transformed cquation. These explanations take no differential coefficients of a function s in regard to x and y, but as dealt with by writing p=-uw, 9=-1/w, and considering account of the possibility of p and q being infinite; this can be variables independent of x, y, s, the too great indefiniteness that might thus appear to be introduced being provided for in another homogeneous equations in u, v, w, with udx+ydy+wdz=o as the way. We notice that if s=(x, y) be a solution of the differ- differential relation necessary for a connectivity; in practice we ential equation, then da=dxdt/dx+dydy/dy; thus if we denote use the ideas associated with such a procedure more often without the cquation by F(x, y, z, p. 9.) =0, and prescribe the condition the appropriate notation. dz=pdx+ady for every solution, any solution such as z = In utilizing these general notions we shall first consider will necessarily be associated with the equations p=de/dx, the theory of characteristic chains, initiated by Cauchy, which q=dz/dy, and will satisfy the equation in its original form. We shows well the nature of the relations implied by the given have previously seen (under Pfafian Expressions) that if five differential equation; the alternative ways of carrying variables x, y, 2, p. 9, otherwise independent, be subject to out the necessary integrations are suggested by con los derecede dz-pdr-qdy=0, they must in fact be subject to at least three sidering the method of Jacobi and Mayer, while a good mutual relations. If we associate with a point (x, y, z) the plane summary is obtained by the formulation in terms of a Pfaffian 2-=P(X-x)+(Y-y) expression. passing through it, where X, Y, Z are current co-ordinates, and equations F =0, G=0, H = 0. If it be a solution in which there is Consider a solution of F=o expressed by the three independent call this association a surface-element; and if two consecutive more than one relation connecting x, y, :, let new variables x'y', 3.7.9* elements of which the point(x+dx, y+dy, +dz)of one lies on the be introduced, as before explained under Pfaffian Explane of the other, for which, that is , the condition dz=pdx+ody pressions, in which is of the form is satisfied, be said to be connected, and an infinity of connected so that the solution becomes of a form ? =4(x+y), s=2-pix,-...-.x.(s = 1 or 2), elements following one another continuously be called a con- \ P = dv/dx', q' = dv/dy', which then will identically satisfy the transnectivity, then our statement is that a connectivity consists of not formed cquations F' =0, Gʻ=0, H'=o. The equation F-0, if y's more than 60% elements, the whole number of clements (x,y,2,2,0) be regarded as fixed, states that the plane 2-5=P'(X-x")+d(Y-y) that are possible being called Co. The solution of an equation point ($+dx", y +d2','+dz%) of the generator of contact being such F(x,y,z, dz/dx, dx/dy) = o is then to be understood to mean finding that in all possible ways, from the elements (x, y, , P. () which satisfy F(x, y, 2, P0=o a set of co? elements forming a con Medyde] (poet + ) nectivity; or, more analytically, finding in all possible ways two Passing in this direction on the surface e' =(*'y) the tangent order. Charac teristle chains, |