UC-NRLF ^B 535 flfl-; Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarytreatiOOairyrich ON PARTIAL DIFFERENTIAL EQUATIONS. PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY IKESS. AN ELEMENTARY TREATISE ON PARTIAL DIFFERENTIAL EQUATIONS laestgntD for tje SSse of StuiJentu in tj^e SJuibcrsitj. GEORGE BIDDELL AIRY, K.C.B., M.A., LL.D., D.C.L. ASTRONOMER ROYAL, FORMERLY FELLOW OF TRINITY COLLEGE. AND LUCASIAN PROFESSOR, SUBSEQUENTLY PLUMIAN PROFESSOR OF ASTRONOMY AND EXPERIMENTAL PHILOSOPHY. IN THE UNIVERSITY OF CAMBRIDGE. ^^-->^:^ 3CC0^IP .EDITIOI) ILonUon : MACMILLAN AND CO. 1873. [All Bights reserved,] /S /Xi-^ ^c, '- ^,?'- 9 \\\ PEEFACE TO THE FIEST EDITION. The work now offered to the University is strictly an Elementary Treatise. No attempt has been made to go into all the varied details, of methods and examples, which present themselves in the wide field of Partial Differential Equations, considered purely as an Algebraical subject. I have endeavoured, however, to omit no important consideration affecting the Principles of those Equations. And I trust that the methods of solution here explained, and the instances exhibited, will be found sufficient for application to nearly all those important problems of Physical Science, which require for their complete in- vestigation the aid of Partial Differential Equations. G. B. AIRY. Royal Observatory, Greenwich, 1866, August 15. PEEFACE TO THE SECOND EDITION. Several verbal alterations are made in this Edition ; two small paragraphs are added ; and some sentences are introduced, referring to works in which .examples of the application of the Theory of Partial Differential Equations may be found. But nothing is changed in the plan of the work, and no alterations are made in the numbering of the Ai'ticles. G. B. AIRY. Royal Observatory, Greenwich, 1873, July 15. INDEX. ABT, PAGE Preliminary Notice on Integration i, i Characteristics of the Solutions of Simple Differential Equa- tions 2f ib. Introduction of the term '* undetermined constant" instead of " arbitrary constant." ...... 3, 2 Introduction of two or more Independent Variables . . 8, 4 Elucidation from Algebraical Geometry , , . . lo, 5 Time is often one variable . , .. . . .13, 7 Treatment of the simplest Partial Differential Equation of the First Order . . . . . , .15, 9 Introduction of Undetermined Function . , . . 19, 11 G-eometrical interpretation of the Solution .... 22, 13 Characteristics of the Solutions of Partial Differential Equa- tions 23, 14 Interpretation of the Solution when one Independent Variable is Time; motion of a wave ..... 24, 15 Other Partial Differential Equations of the First Order . . 25, 16 Virtual Identity of Different Solutions . . . . 27, 18 Treatment of the simplest Partial Differential Equation of the Second Order ....... 30, 21 Geometrical Interpretation of the Solution . • . . 3I> ib. Treatment of another Partial Differential Equation of the Second Order : First, by Change of Independent Variables . . 35, 24 Second, by Separation of the Symbols of Operation from those of Quantity . . , . • 37> "27 Other Partial Differential Equations of the Second Order . 41, 30 Equations whose solution is desired 44> 33 vni INDEX. ABT. PAGE Further consideration of the Separation of the Symbols of Operation from those of Quantity . . . . 45, 34 Treatment of Partial Differential Equations when the solution is limited by pre-arranged conditions . . . . 49, 40 Final Determination of the Functions which are undetermined in the general solution 53, 42 First Example. Tide-wave in interrupted channel 54, i6. Second Example. Vibrations of musical strings . 57, 46 Considerations on the necessity of connexion between the number of undetermined constants or functions and the order of the equation 64, 55 Figure i^ Diagrams') „ 3f At the end Stereoscopic cards of the four diagrams . . In the binding LIBKAK Y UNfYEKSlTY <>f ^OALIFOKNLV. ON PARhlL DIFFERENCIAL EQUATIONS. PRELIMINARY NOTICE ON INTEGRATION. L In all that follows, we shall suppose that it is always possible to effect simple integration ; inasmuch as any difficulties of integration, connected with the solutions of Partial Differential Equations, do not affect the principle of those solutions. Thus, for instance, we shall not hesi- tate to represent an unknown function of a; by x {x), (the second differential-coefficient of %(a;)), on the assump- tion that, whatever be the form of p^'' (ir), we can in some way find the function ^ (x) of which it is the second differ- ential-coefficient. CHARACTERISTICS OF THE SOLUTIONS OF SIMPLE DIFFERENTIAL EQUATIONS. % Before entering on the subject of Partial Differ- ential Equations, it may be convenient to consider some of the characteristics of the solutions of Simple Differential;, . Equations. 8. To begin with Simple Differential Equations of the first order. Suppose, for facility of geometrical illus- tration, we consider the equation y -^ = ay or the equation D. E. B ^^ 2 ON PARTIAL DIFFERENTIAL EQUATIONS. dx y 7r~^' of which the former, translated geometrically, in- dicates that the subnormal of a plane-curve (to be found) is constant, and the latter indicates that the subtangent is constant. The algebraical solutions are easily found : in each, there is a constant, which does not occur in the original differential equation, and is not defined by it ; a constant of that class described (perhaps improperly) by the term "arbitrary," but which really means "not yet determined, but enabling us by proper determination of its value so to fix the value of x corresponding to a given value of y that we can adjust the solution to some specific condition." The reader is requested to observe that instead of the term " arbitrary constant," we shall always use the term " undetermined constant.'' 4. If we treat the geometrical translation of the dif- ferential equation by a geometrical process, always draw- ing a normal or a tangent (as the case may be) so as to make the subnormal or subtangent constant, then drawing by means of it a small portion of the curve, then repeating the process, &c., we may produce a polygon which will approach to the strict solution, with smaller errors (by taking the sides small enough) than any small quantity that can be assigned. In each case, however, a starting-point is necessary. 5. In both ways (the algebraical and the geometrical) of treating the problems these conditions manifestly hold :_ CHARACTERISTICS OF SIMPLE DIFFERENTIAL EQUATIONS. 3 The curve, in each case, is one definite curve. The curve, in each case, is a continuous curve, ex- pressed by the same equation through its whole extent. Even if there be isolated points or curves, still the same one equation defines the whole* It is necessary to introduce one undetermined quantity, enabling us to adjust the curve to a specific con- dition given by special considerations : that unde- termined quantity is however a simple constant. 6* Let us now consider a Simple Differential Equation of the second order : such, for instance, as is "given by this problem, " To find the curve in which the radius of curva- ture is a function, given in form, of the ordinate." Here the algebraic solution gives a formula requiring two unde- termined quantities, still simple constants : the equivalent geometrical treatment shows that we require two elements^ (as, for instance, the value of x and the inclination of the tangent, for some one value of y). But all the conditions hold which are mentioned in Article 5 : the only difference being that, instead of one undetermined quantity, a simple constant, there must now be two undetermined quantities, simple constants. 7. The conditions which will be found to hold in the solutions of Partial Differential Equations differ very re- markably from these. B2 4 ON PARTIAL DIFFERENTIAL EQUATIONS, INTRODUCTION OF TWO OR MORE INDEPENDENT VARIABLES. 8. It is convenient, in Simple Differential Equations, to contemplate one quantity z as expressed algebraically in terms of another quantity x (whether it be actually so expressed, or not) : and to consider its differential-coeffi- cients dz d [dz\ d^z ^ dx ' dx \dxj dx^ ^ '' aa being formed by taking the limiting values of fractions, in each of which the denominator is an increment in the value of X) and the numerator is the corresponding incre- dz ment of z, or of -y- , &c. And this is expressed by saying that X is the " independent variable." Geometrically it is illustrated by supposing that we consider an ordinate z of a plane curve, as also the inclination-tangent of the curve's-tangent, the change of that inclination-tangent for a small change in the value of x, &c., to be expressed, in terms of the abscissa x. 9. There is no difficulty in conceiving z to depend on two quantities x and y, combined in any way and with any constants under any functional formula; and in considering that we may at pleasure change the value of x without changing that of y, or may change the value of i/- without changing that of x, or may change both simultaneously. If we do not vary y, y is pro tempore^ a constant : and by, uZ CbZ varying x alone, we may form — , j-^, &c., just as in the TWO VARIABLES IN SOLID GEOMETRY. 5 functions which we have illustrated by reference to a plane curve. If we do not vary x, then x is "pro Umpore a con- stant, and by varying y alone, we may form -p, -j-^-, &c. But if we vary both x and y, then we have the two series of diflferential-coefficients which we have just set down, and also -7 ^ or -^ 7- (which, as is known in the ex- dx . ay ay . dx ^ pansions obtained by the Differential Calculus, have the same value), with other coefficients of analogous form, which do not appear in the succession of coefficients, such as those of Article 8. Here we consider x and y as " two independent variables," which are strictly independent of each other : and ^ as a function of both. 10. Algebraical Geometry of three dimensions assists greatly in illustrating these algebraical conceptions. If x, y, z be three rectangular co-ordinates, then the expression oi z by means of x and y determines the numerical value of the height of an ordinate z that must be erected over the point on the plane xy, which is defined by any nume- rical values of x and y, in order that the elevation of the summit of that ordinate z may represent the value of our algebraical function. If we do this for an infinite number of values of x and y, we determine an infinite number of . ordinate-summits, all which lie in one curved-surface. If, supposing this curved-surface formed, we then contem- plate the values of z with x invariable and y variable, we shall include all these values which are in a plane parallel ^6 ON PARTIAL DIFFERENTIAL EQUATIONS. to the plane yz^ and at the given distance x from the origin of co-ordinates. This is the same as forming a sec- tion of the curved-surface by a plane parallel to the plane yz, and considering the curved-intersection as an ordinary dz d^z plane-curve. Thus we can obtain -7- , -^ , &c. as in Ar- ticle 8 : but it must be remembered that, in general, x is lurking in their expressions as a constant ; and therefore, in the complete differentiation (with respect to x) of any , formula in which they may occur, each must be considered as a function both of x and of y. Similarly, the values of z with y invariable and x variable will be formed by a sec- tion of the curved-surface, made by a plane parallel to xz : dz d^z the differential-coefficients -^ , -^2 > &c- can be formed ; cLx dx and a similar caution with regard to their dependence both on X and on y must be borne in mind. 11. The coefficients -^ , -y-^, &c. are, as in Article 8, to be considered as defining the inclination-tangent of the curve' s-tangent, the change of that inclination-tangent for a small change in the value of x, &c., &c., in the curve formed by the intersection of the plane parallel to xz with the curved-surface. In like manner, the coefficients 1- , -r-^ , &c. define similar elements in the curve formed dy dy' by the intersection of the plane parallel to yz with the d^z curved-surface. But the coefficient -, j- requires further dx .dy ^ TIME IS OFTEN ONE VAKIABLE. ( -^ 1 . Now -4- de- explanation. Its real expression is -7- ( -r- J • Now -7- fines the inclination-tangent of the curve's-tangent in the sectional plane parallel to yz : and therefore its differential- coefficient with respect to x defines the rate at which that inclination-tangent, in the plane parallel to yz, is changed by shifting the sectional-plane in the direction of x. Analogous explanations apply to analogous succeeding terms. 12. Algebraically, there is no difficulty in conceiving ^ as a function of any number of independent variables, as u, V, w, X, y, and in treating the viarious differential co- efficients, which may rise to any degree of complexity. But we cannot extend any further the elucidation derived from Solid Geometry. 13. Although, in strictness, no difference is really made by any physical peculiarity in the nature of the quantities represented by the independent variables, pro- vided that their magnitudes can be defined by numbers and can therefore be treated by algebraical process ; yet it may be well to mention that, in some of the most important applications of Partial Differentials, one of the independent variables is the expression for time. Thus, suppose that we are considering the nature of the disturbance which each particle of air is undergoing in a musical pipe : we want to know the entire movements of the particle whose original ordinate was X; and this implies that, in the general formula for the disturbance, x (the general symbol of S ON PARTIAL DIFFERENTIAL EQUATIONS. X) is to have the special value for that particle, and i^ therefore to be made constant for that particle, but that no limit is to be put on the variations of L We also want to know the state of disturbance of all the particles at a certain time T, and this implies that t is then to be made constant, but that x is to admit of all possible variation. And these can only be comprehended in a general formula which admits of variations both of x and of t Similarly for the motion of the particles of waves of water, &c., &c. 14. In Simple Differential Equations, the data of the problem, whether mechanical or geometrical, usually dz (Pz lead to an equation between x, z, -^ , -y-^ , &c., from which we desire to obtain a general expression for z in terms of x. Similarly, in Partial Differential Equations the data of the problem usually lead to an equation be- dz dz d^z d^z d^z ^ „ ... we desire to obtain a general expression for z in terms of X and y. It is the object of the present treatise to shew how this may be done, and how the meaning of the solu- tions may be explained, in some of the simpler cases : and we now proceed with the Solution of Partial Differential Equations of the First Order, TREATMENT OF EQUATIONS OF THE FIRST ORDER. 9 ' PARTIAL DIFE TREATMENT OF THE SIMPLEST PARTIAL DIFFERENTIAL EQUATION OF THE FIRST ORDER. 15. The simplest Partial Differential Equation of the first order is dz dz dz dz ^ ax ay ax ay [Here, and in all subsequent operations, we shall put u for ax + y.] Evidently this equation is satisfied by making z=^u. For -j-= ay ^- = 1, and therefore -, a -^ = 0. But ax dy dx dy it is also satisfied by making z = ^{u), where ^ expresses a function whose form is totally unlimited. For then dz ,. . du ,r, . dz ., , . du, .f,. ^ ^ = <^W.^- = .^(«)xl; and therefore 16. We have thus found a solution of considerable generality in form ; but there is no evidence that the generality is complete. In order to obtain complete and certain generality, we shall use a process of Change of Independent Variable, guiding ourselves in the selection of a new Independent Variable by the indications derived from the last imperfect process. 10 ON PARTIAL DIFFERENTIAL EQUATIONS. 17. Instead of considering ^ as a function of x and y, let US consider ;^ as a function of x and u. [Such a sup- position is certainly competent to represent z, because, if we had z expressed by x and y, we have merely to sub- stitute for 7/ its value u — ax, and z then becomes expressed by X and u.] Having z then a function of x and u, where u is itself a function of x and y, we proceed to express the original equation in terms of the new differential-coeffi- cients. We shall put -y-^ for the entire differential-co- efficient in regard to every way in which x could appear dz when 00 and y were used, and -j- for the differential-coeffi- cient when x and u are nsed: the original equation is therefore to be written dx dy 18. ^ is a function of x and u^ where u depends on X and y. Therefore d{z) _ dz dz du __ dz dz dx dx du ' dx dx du ' and dz __ dz du ^ dz ^ dy du ' dy du Substituting these in the equation which is to be solved, _ d(z) ^ dz _ dz dx dy dx* INTRODUCTION OF UNDETERMINED FUNCTION. 11 That is to say, ivhen z is expressed in terms of x and u, ax The integration of this simple quantity involves the most important considerations of the whole theory. 19. By the ordinary rules of integration, z = G, where C is something which does not contain x. But what may it contain ? It may contain every thing whatever except x. It may contain any constants whatever. It may contain u : for the differentiation with respect to x will not touch u, which is another Independent Variable and is a constant with respect to x. It may contain any function of u. It may contain these included in the form " any function of constants and u" But, in ordinary algebraical language, this would be sufficiently described as " any function of %' provided that we bear in mind that ii may be combined with any con- stants, and may even actually disappear, so as to leave constants only. This being understood, the solution of the equation is z = , or - . The same applies to every other ordinate z. And thus it appears that the whole curve defined by the summits oi z maybe conceived as changing its shape by sliding backwards with a definite velocity -. This is the characteristic of a wave, in which (as in waves of water, to take the most familiar instance) the form travels while the particles do not travel with the form. Whenever therefore we meet with such a term as ^ {ax-\-y) we may at once take for granted that one of its applica- tions may be to the motion of a wave. If the value of z had been 'y^ibt— ex), we should have found in like manner that the wave is travelling forwards. These remarks do not in any degree interfere with those of Article 23. OTHER PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 25. We proceed with the equation -^ — ^-f = <^{^i y)^ where the form of the function a is given. We shall solve it by the method of Change of Independent Variables. Let u = ax-\-y, v^ex +fy, where e is not equal to of, (which, by giving different values to e and /, includes the three changes in Articles 19, 20, 21). The value of x in terms of u and v is'^. , that of y is — ^^ . Treat- MORE GENERAL EQUATION OF FIRST ORDER. 17 ing the equation as in Article 21, we find whence, integrating on the same principles as before, , , . , 1 f (fu — v —eu+av\ /),/ where, in the integration with respect to v, u is to be con- sidered constant ; and, after the integration, ax -{• 7/ and ex -\-fy are to be substituted for n and v, 26. On trying this process upon any ordinary function (as a function in integral powers of x and y, or a circular or exponential function) of which the integrations are easy, it will be found that e and / disappear from the result. It may be well to shew this in a simple instance. Let one term of a {x, y) be x^ . y^, y >. % > { ^y ^^ The corresponding term of the integral ip /^ /- y -1 r V^ % Integrate by parts, beginning with the first factor; iiA,> gives \ (p + 1) . {af- e) D. E. C 18 ON PARTIAL DIFFERENTIAL EQUATIONS. and, by repeating the process, we shall obtain the result in q-\-l terms, in powers of x and y, and without e and /. 27. If we had begun the integration-by- parts with integration of the second factor, we should have obtained a different series of terms of powers of x and y ; yet the difference would have been such as to produce no difference in the final result, as applicable to any physical investiga- tion in which the undetermined function is to be adapted to given physical circumstances. A simple instance will shew this. Let a(x,y) = x . y^. Then we obtain the two following series for the integral : First, kiecond, z = ir{ax-\-y)'^j^,{-4,axf-y']: The second value, it is easily seen, may be put under the form ■ 1 ■ V -^ {ax yy) + —-2 {6aV2/' + 4aVy 4- aV} - j-J^ {/ + 4^ir2/' 4- 6aV/ + 4aV2/ + aV} VIRTUAL IDENTITY OF DIFFERENT SOLUTIONS. 19 = '>|r (a^ + y) - ^j-,^ {ax + ?/)* + j|^ [6a'xy + 4aVy + aV}. Now — Y^r-o {ax + 2/)* is a function of {ax 4- y) ; and, when we proceed to determine the form of the function -v/r {ax+y), which must be adopted, in order, with the other terms of the expression, to satisfy any condition prescribed by physical considerations, we must at the same time taka into account — — -^^ {ax + T/Y as a function of {ax + y). We x^a may therefore at once unite it with '\jr {ax+y). Call their sum X (^-^ + y)' Then the second solution is « = X («a^ + 3/) + 1^. (6aW + 4aVy + a^x') ; precisely the same as the first, except that the symbol % stands in the place of 0. But in both cases the form of the function ({> or ;j^ will be determined by the consideration of satisfying the same physical condition. And therefore, whether we write (j> or ^, we shall afterwards arrive at the same expressions; and therefore the two solutions, first and second, are in use identical. 28. In many cases, when the form of solution has been obtained by the process of Article 25, the details will be obtained most readily by assuming the form of solution with indeterminate coefficients. c2 20 ON PAETIAL DIFFERENTIAL EQUATIONS, 29. It is not itttended here to go into any discussion of the more complicated forms of Partial Differential Equa- tions of the first order. The following however may be mentioned as flowing immediately from the solutions which we have found. and the equation is immediately reduced to the form already treated. If the first side is e{x) x-^ ctx ^{y) x -j^ , let^ = ,'(-) and ^^ = ^(y), and the equation is reduced to the saijie form. ^^i"^^ ^ ""'^"^^^ ^^^^' ^ ^-/t(^); then dz^ ^ dz f , . dz^ __ dz dx '^ dx '' dy '^ dy' and the equation becomes ax ay In reference to physical investigations, the theory of Partial Differential Equations of the first order is princi- pally valuable as introductory to that of the second order. In this view^ it is hoped that the instances here given are suflS.cient. SIMPLEST EQUATION OF THE SECOND OBDER. 21 TREATMENT OF THE SIMPLEST PARTIAL DIFFERENTIAL EQUATION OF THE SECOND ORDER. SO. Tke simplest Partial Differential Equation of the second order is — — — = 0. dx . dy Integrating with respect to x, and remarking that, by virtue of the reasoning in Article 19, there must be added to the integral an undetermined function of y. Integrating this with respect to y, and remarking that, for the same reasons, an undetermined function of x must be added, z = (j>{y)+y}r ix). GEOMETRICAL INTERPRETATION OF THE SOLUTION. 81. This equation and its solution admit of easy geometrical illustration. Referring to Article 11, it will be dz . seen that -j- is the trigonometrical-tangent of the angle between y and the curve-tangent, in the curve formed by the intersection of a curved-surfacewith a plane parallel to zy. And the equation -7 — -y- = ^j ^^ 3~ ( 3" ) =" ^y denotes dz that, when x is changed, -j- undergoes no change ; or that 22 ON PARTIAL DIFFERENTIAL EQUATIONS. the inclination in the plane zy will be the same for a point more or less advanced in the direction of x as for the point under consideration. The curve-tangent may be absolutely higher or lower in the direction of 2, but it will preserve the same inclination to y. As this applies to every point of the intersection-curve, it follows that the intersection-curve more or less advanced in the direction of X will have the same form as that at the point under consideration ; or that all sections by planes parallel to zy give the same curve, though perhaps at different elevations. And from this it follows (considering that, when two similar curves are separated in the direction of x, the slope from every point of one to the corresponding point of the other, in the direction of x, must be the same) that all sections by planes parallel to zx give the same curve, but not necessa- rily the same as those given by planes parallel to zy. Then the elevation of any point z may be considered as composed of these two parts ; first, the elevation of the corresponding point of that zy curve whose plane passes through the origin, which elevation (since the form of the curve is the same for all values of x) may be called ^ (?/); secondly, the elevation of the point z above the point last considered' which elevation (since the points are connected by a curve whose form is independent of y) may be called a^ {x). The whole elevation of the point z will therefore be (y) + ylr(x). 32. These functions may be discontinuous, for the reasons stated in Article 22. GEOMETRICAL INTERPRETATION OF THE SOLUTION. 23 33. In figure 2 is a representation of one of the solids in which (f) {y) and -v^ {x) are both continuous functions. It is scarcely necessary to remark that the forms of such solids may be infinitely varied ; thus (taking the verl. x as 2 2 origin of coordinates), if ^ = ^ / ' ^^ both sections be parabolas with different parameters, the curve-surface is a paraboloid with elliptic base; if ^ = : — = — , the curve- surface is the paraboloid of revolution (from which, figure 2 2 2 was drawn) ; if ^ = j- , the curve-surface is like a ah limited portion of the interior of an annulus, or like a mountain-pass, &c. &c. &c. In figure 3 is a representation of a curve-surface in which one function is continuous (the sections being similar parabolas), the other is discontinuous (each section being two sides of a triangle, the triangle being the same throughout). . In figure 4 is a representation of a surface in which both functions are discontinuous (each section being two sides of a triangle, but the triangles in the plane xz being different from those in the plane yz). It forms, in fact, a pyramid with trapezoidal base ; two vertices of the trape- zium having the same value oi x, and two having the same value of y, and the vertex of the pyramid being above the intersection of the diameters of the trapezium. 24 ON PAKTIAL DIFFERENTIAL EQUATIONS. 84. The solution of the equation -^ — -^ = a {x, y) obviously is, TREATMENT OF ANOTHER PARTIAL DIFFERENTIAL EQUA- TION OF THE SECOND ORDER. FIRST, BY CHANGE OF INDEPENDENT VARIABLES. 85. To solve the equation j-^ — a^ — : = a {x, y). [This equation is the most important of all, especially in reference to those physical theories in which wave- transmission, of sound, or of light, or of water, &c., is explained by mechanical theory.] The best method of solving this equation is by the Change of Independent Variables. Let n^ax + y, v = ax — y, ^which giveaj= -^^,2/ =-^ j . Consider 2? as a function of x and y because it is a function of u and v. Then dz dz du dz dv dz dz ax du ax dv ax du dv d^z __ d fdz\ _ d fdz\ du d fdz\ dv dix? dx \dxj du \dx) dx dv \dx) dx SOLUTION BY CHANGE OF VARIABLES. 25 \ d (dz dz \ J dv \du dv J d (dz dz \ d [dz dz au \du dv dw da . dv dv dz _^ dz da dz dv __ dz dz ^ dy da dy dv dy da dv ^ d?z _ d fdz\ _ d fdz\ du d (dz\ dv dy^ dy \dy) da \dyj dy dv \dyj dy d (dz ^ dz \. ^ d (dz ^ dz ^\ ^ da \du dv J dv \du dv J _ d'^z d^z d^z d\j^ da . dv dv^ ' T-r d^z od^z , 9 d^z Hence -^-7 —a -^-o = 4(2 - or da;^ dy^ da , dv' And the original equation, divided by 4a^ becomes d^z __ 1 /u +v u —v\ Whence, by Article 34, ,/N ./N ir [ (u +v u -v\ »/ N ./ \ Iff (u-^-V U-'V\ 26 ON PARTIAL DIFFERENTIAL EQUATIONS. where, after the integration, for u and v are to be put ax + y and ax — y. 35* If the right-hand term of the given equation were a simple function of x, multiplied only by constants and the first power of y, as if let 2Jj = ^ — (S 4- cy) x ^ (x) : and the equation becomes (X> Z^ 2 ^ ^\ f\ of which the solution (by the formula above, when a = 0) is Zj^' = (j) (ax — y) + '^ {ax +y) ; or z — ^ [ax - y) +'^ {ax + y) + {h -{-cy) x ^ {x). .85**. When a = 0, the solution z = ^{u) +'\Jr {v) may be represented geometrically in the same manner as z = (p {x) + yjr (y) in Article 33, with this difference only, that u and v are ordinates on the plane xy, which are not at right angles, except in the case when a = 1. 36. When one of the independent variables is time, it will be seen by the same reasoning as in Article 24 that the two undetermined functions represent two waves travelling in opposite directions. .SOLUTION BY SEPARATION OF SYMBOLS. 27 SECOND TREATMENT OF THE SAME EQUATION, BY SEPARA- TION OF THE SYMBOLS OF OPERATION FROM THOSE OF QUANTITY. 87. A second method of solving the equation d^^z ^d'^z — a dd(? dy^ a {x, y) is founded upon the " separation of the symbols of opera- tion from those of quantity." This theory is, in fact, merely a convenient form for exhibiting the indubitable results of legitimate algebra ; but it sometimes serves to suggest new methods of treating equations, which, when verified, are useful. 38. The equation dz dz .^ ^^. may be written, almost without departure from ordinary notation, the connexion of the left-hand bracket with z which follows it being, however, not by multiplication but by differen- tiation. And, if- 28 ON PARTIAL DIFFERENTIAL EQUATIONS. we may, on the same principle, write it and therefore still on the understanding that the connexions on the left hand are not by multiplication but by differentiation. 39. Now, if we treated all the left-hand symbols as signs not of operation but of quantity, their product would be or, pursuing the fanciful idea still further, d^'^'^'df' and this suggests the idea that, if we follow up the true differential operations instead of the fanciful algebraical multiplications, we shall arrive at that result in its true differential meaning. And so, in fact, it proves. For, by actual differentiation, d fdz dz\ ^ d^z d'^z dx \dx dy) "" dx^ dx . dy^ dy) dx^ dx . dy^ d_z\ ^ _^ d'z dy \dx ^ "* dyj "~ dx . dy dy^ ' d fdz dz\ d^z ^d'z SOLUTION BY SEPARATION OF SYMBOLS. 29 and, adding the two lines, d d\. (dz dz\ d'^z „ d^z ■\-a~r] = -n-a; \dx ay) \dx ' "^ dy) dx^ ^ dy^ Here the analogy with the algebraical operation has led to a real gain of convenience, by shewing that we can here break up the operation for a Partial Differential Equation of the second order into two operations for the first order. 40. Thus, to solve the equation d^z ^d z . . we have first to solve the equation '-^-^ - «(-'. from which (X, Y) will be found ; and then to solve the equation 1+4; -<^-'')' from which z will be found. Both operations are effected by the process of Article 25; the most convenient assumption however being u ^ ax-k-y^ V = ax--y» We shall return hereafter to the " separation of symbols.^* 30 ON PARTIAL DIFFERENTIAL EQUATIONS. OTHER PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 41. To solve the equation or d^z , I — --.' d'^z ^ ^,-(aV-ir.^. = 0. It does not appear possible to solve this equation gene- rally, except by the use of imaginary symbols. If we take the second form of the equation (as written above), and apply the solution of Articles 35, &c. If, as is usual, we suppose <^ and i/r to represent real functions, it will be impossible to destroy the imaginary terms in this expression except by making -i/r the same as 84 ON PARTIAL DIFFERENTIAL EQUATIONS. which we have not succeeded in integrating generally in a finite form. In investigating the radial motion of a spherical wave of air whose center is the center of the earth, supposing the elasticity to be as the rfi" power of the density (where n must be greater than a certain quantity which is > 1 , in order to make the whole mass of atmosphere finite), the resulting equation takes this form : I r )dr\dr rj r^\dr r) ,.d^R 4a^R ^ df r^ This equation may be much simplified, but in any form we have not rendered it inteofrable. FURTHER CONSIDERATION OF THE SEPARATION OF THE SYMBOLS OF OPERATION FROM THOSE OF QUANTITY. 45. In Article 39 attention was called to the method of solving the equation then under consideration by the ''Separation of the symbols of operation from those of quantity." This principle may be applied to an extensive system of equations, which are however, at present, matters rather of curiosity than of physical value. It may be sufficient here to indicate one class. SEPARATION OF SYMBOLS. 35 If the equation ^-^+^^-^+^rf^-w^-*- +^^^==«(^'^) can be expressed as (-^i a-r] X f-T- — i-T-)x &c. to n terms x ir = a (a;, y), \dx dy) \dx dy) ^ '^^* then the process of Article 40 can be applied with so little difficulty that it appears unnecessary for us to delay further on it. In the successive changes of Independent Variable, it will be found convenient to take the factors in successive pairs ; thus, for the effect of the first pair of factors, where assume ax-\-y=^Uy hx-^-y^v, and proceed as in Article 25. 46. There is one exceptional case^ however, well brought to notice by this method of treatment, which merits further attention ; namely, that in which two factors a, 5, are equal. To take the simplest case, suppose d?z . dh ^d^z d2 36: ON PARTIAL DIFFERENTIAL EQUATIONS. \dx dy) \dx dyl Here we may assume ax^-y = w, ex-\'fy =^ v, e and/ being any constants whatever whose proportion is not a : 1 ; the value being admissible for either. Then dx du dv ^ d{X, Y) ^ d(X,Y)^^^ ^ d(X,Y) ^ dy du dv '^ ^ and, putting {X, Y) for (^-«x,) ^> the equation ^^ - a^j (X, Y) = 0, becomes (^""^/) ^ — = ^j whence (X, Y) = (f> {u). And, putting for (X, Y) its value, (^-"|)^ = '^(^)' or, by the same substitution which was made in regard to the differential coefficients of (X, Y), SEPARATION OF SYMBOLS. S7 integrating this, and remarking that ^ {u) possesses the properties of a constant as regards integration with respect to V, (e — af) z = V ,(f> (u) +'\}r (u); or Observing that a constant factor may be supposed to be included in the form of an undetermined function, we may- express z in either of the forms, ^x^2 iy +«^) +^2 (y +«'^) ; all which, as applicable to the satisfaction of any assigned conditions, will be found to be identical, on employing the reasoning of Article 27. 47. In pursuing this subject, the signs / and d may be considered as reciprocal, or / = d"\ In some cases, equa- tions may be varied by carrying symbols, such as ±_ d_ dx dy^ which hold the position of multiplier on one side of the equation, to the position of divisor on the other side, or 38 ON PAKTIAL DIFFERENTIAL EQUATIONS, as multiplier in the form \dx dy) And in this shape they may be substituted in abstruse and general theorems. For instance, to take a case which (in this theory) is almost elementary, we know that If 7^ = — 1, this becomes, reversing the sides of the equation, or [ -^^ — h « J u = e"""^ I u e""". But the equation dz dz __ dx dy~~ * d or (d . d\ \dx ay] may, in accordance with these principles, be put in the form \dx dy) ' SEPARATION OF SYMBOLS. 39 and the second side agrees with the first side above, provided that e-j- he put for a and treated as a constant. Then we shall have Z=:€ dy \ d U e dy J a solution to which, in some cases, a real meaning may be given. 48. This principle, as a purely algebraical and symbolical process, possesses very great power, and leads to very remarkable results. But the reader cannot fail to observe that it carries with it no evidence whatever for the validity of results (such as is conveyed by the opera- tions of quantitative algebra, or by the steps, properly pursued, of the differential calculus), for which it must rely on subsequent veriti cations. As aiding the appli- cation of Partial Differential Equations to physical investigations it possesses little value. The further examination of it would therefore be out of place in this Treatise. The student who desires to follow it up will find much information in Boole's Treatise on Differ- ential Equations, Gregory's Examples, and similar works. 40 ON PARTIAL DIFFERENTIAL EQUATIONS. TREATMENT OF PARTIAL DIFFERENTIAL EQUATIONS WHEN THE SOLUTION IS LIMITED BY PREARRANGED CON- DITIONS. 49. We have seen in Article 44 that there are some Partial Differential Equations of which the solutions cannot be obtained by known methods, so long as the utmost generality is required in the solutions. But in some cases, by attaching limiting conditions to the solu- tions, the same equations may be integrated with ease; as will be seen in the following instances. 50. In considering the tidal disturbance^of water in an equatoreal canal round the earth, retarded by friction pro- portional to the velocity (see the EmyclopcBdia Metropolis tana, Article Tide& and Waves), we arrive at the following equation {x the original ordinate of a particle measured along the canal, X the horizontal disturbance of the par- ticle) : d'X rr-r'. ^ A^ 2cJ:'X —77^- = ±1 sm (it —mx) —f -^ — ha — r-rr ; where H, /, a\ are constants depending on the moon's attraction, the coefficient of friction, and the depth of the sea. This equation cannot be solved generally. But, remarking that the only part of the solution which has any interest for us is that which follows the same law of pe- riodicity as the lunar motions, we may assume X = A sin [it —mx) +^cos {it —mx)^ SOLUTION UNDER LIMITING CONDITIONS. 41 wbere A and B are constants to be determined. On substituting this, the values of ^ and B and the expression for X are found without difficulty. 51. Before adverting to a specific solution of the equation -^—^ -\-a^ -r-^^ = 0, we will make the following remark on the general solution. When an exponential form is adopted for the functions in the general solution, Article 41, the general form may be retained conveniently: but in other cases it is usually necessary to expand the functions. Performing this process according to the ordi- nary algebraical rules, the general solution becomes -&c. ax which, as will be found on trial, satisfies the equation. A solution by series, however, can rarely be considered as quite sufficient. 52. In considering the motion of ordinary small waves in a sea of uniform depth {x horizontal in the direction in which the wave is going, y vertical measured upwards from the bottom), we find (see Tides and Waves) cPX dlX_r. df ^ dx^ " The cumbrous form of the general solution of this equation would make it almost useless to attempt to apply 42 ON PARTIAL DIFFERENTIAL EQUATIONS. it to the special case. But the nature of the case permits us to assume X = iZcos {nt—mx) +>Ssin {nt —mcc), where R and S are functions of y. Substituting, we have whence and, after some reductions peculiar to the problem, X = A {€^^-{- €'""') cos {nt-mx-B). FINAL DETERMINATION OF THE FUNCTIONS WHICH ARE ' UNDETERMINED IN THE GENERAL SOLUTION. 53. In nearly the whole of these solutions, our re- sulting expression for the quantity which it is our object to find is accompanied with undetermined functions. It is now an object of great importance to determine those func- tions. Although in reality the process for doing this may be stated in brief rules, yet it will be better understood from an exhibition of its application in one or two actual instances than in any other way. 54. Example 1. In the consideration of the tidal motions of water without friction in an equatoreal channel round the earth, X being the horizontal displacement at INVESTIGATION OF THE UNDETERMINED FUNCTIONS. 43 any time ^ of a particle whose ordinate measured round the channel from a fixed origin was Xy the equation is -jj^ = H sin [it —mx) +c -7-y ; where it is double the moon's hour-angle from the origin, and mx is double the longitude-angle of the point x from the same origin; and where c^ is a constant depending on the depth of the water. This equation can be com- pletely solved : its result is IT llv — t which we shall write (7 sin (it-mx) +(f> {ct+x) +'\fr (ct—x). Each of the three terms represents a wave, but they are waves of different characters. The first wave corre- sponds in its velocity with the lunar force ZTsin (it—mx) which produces it : for this the author has introduced the term /orced! wave; it depends entirely on the lunar force, and would not exist if there were no lunar force. The second wave is travelling in the direction opposite to the moon s apparent diurnal motion, and the third wave in the same direction as the moons apparent diurnal motion; for these the author has introduced the term free waves; their velocity is independent of the moon's motion; the two waves, or either of them, may or may not exist ; they are entirely independent of the moon's action, and the 44 ON PARTIAL DIFFERENTIAL EQUATIONS. question of their existence is generally independent of the moon s action and of the forced wave. This is the state of thinofs when there is no material limit to the lens^th of the canal. 55. But suppose that the canal is limited by the ob- stacle of land at its two ends (as in the Mediterranean Sea, all minor gulfs, &c., being put out of consideration). Then the forced wave alone is not sufficient for a solution. For, the existence of the terminal obstacles (whose ordinates we will call a and b) requires that, at those obstacles, the hori- zontal displacement of the water be nothing at all times ; and therefore when x=^a or x = b, X must = whatever be the value of t. Now this cannot hold with the forced wave alone ; for, the values which the first term assumes for a? = a and x = b, namely, (7 sin {it —ma) and Csin {it -^mb), will not vanish for all values of t. We must therefore have recourse to the two undetermined functions; or, in other words, we must now necessarily introduce the two free waves as an indis- pensable part of the solution : and we must determine their elements so that the two conditions, of X always = when x = a and when x — b, shall be satisfied. 56. The term expressing the forced wave is a simple periodical term having i;^ in its argument. Therefore, in order to destroy this at all times for definite values of x^ the free waves must be expressed by simple periodical terms having it m their arguments. Therefore the two UNDETERMINED FUNCTIONS IN TIDAL PROBLEM. 45 undetermined terms ^ {ct +ic) and ^/r [ct — x) must have the forms A^ sin 1^ {ct ^x) \ +jB^ cos |- (c^ -\-x) I , and ^2 sin \- {ct^x) h ^jB^cos j- (ct—x) [ ) and the entire expression for X will have the form Cmiiit-^mx) - A^ sin (i^ -f- ^) +5, cos (zif -\--x) + ^jj sin {it — x) +-S2 cos {it — x) \ or, expanding the trigonometrical terms, and putting e for A,-yA^, /for A^ - ^,, ^ for B^ +^,, /i for ^, ^B,, X = ^ sin it X ■! (7 cos two; 4e cos — + A sin — ^ 1 c cj + cos i^ X I — (7 sin mx +/sin yg cos — ^ . Making this = when i» = a and when £c = 5, and writing a for — and /3 for — , we have the four equations, c c = (7 cos ma 4-^ cos a +^ sin a, = — (7sin7wa+/sina +^cosa, 46 ON PARTIAL DIFFERENTIAL EQUATIONS. = (7 COS mb 4- ecos ^ -\-h sin 13, = — Gs'mmh -h/sin ^ +g cos /3. Solving these equations, c sin (y8 — a) = C (— cos ma . sin /3 +cos m5 . sin a), /sin(/3— a) = C (— sin??ia. cosy8+sin m&.cos a), 5rsin(/3— a) = C ( sin ma . sin /3 —sin mS . sin a), A sin (/3 —a) = C(cosma. cosyS -cos7n6.cosa). Substituting, and restoring the original notation, ^ H , .. . H ^ — V"! ^ sin u^ —mx) -: r X (cm — ^ ) sin j- (6 — a)h I sin (iit— ma). sin \-(b —x) [ +sin(/i^ — m/.>),sin \- (x —a)[ . Thus the terms, undetermined in the general solution, are now completely determined so as to satisfy the special conditions of the problem. The entire expression for X, it will be seen, satisfies the original partial differential equa- tion, and .also makes X = at all times when X = a and X == b. ' 57. Example 2. In the ordinary problem of vibrating musical strings, wKere no force is supposed to act after the string has been put in motion, x being measured longi- tudinally along the string and z being the small transversal UNDETERMINED FUNCTIONS IN MUSICAL PROBLEM. 47 displacement of any point (see Sound, Article 73), the equation for z is where o? = gL, L being that length of the same string whose weight will represent the tension of the string : and its solution is z = (j> {at -\-x) -i-ylr {at —x). Here the solution is expressed entirely by the two undetermined functions. If the initial circumstances of disturbance (namely, the displacement and the velocity of every point) were given, those functions could be deter- mined, as will be shewn below. But even without an absolute knowledge of the forms of the functions, some very important properties of the solution may be ascer- tained thus. 58. Let the ordinate of the near end of the string be 0, and that of the further end l. Then, whatever be the value of ^, 5J is when a; = or when x = L That is, ^ {at) +f {at) ^ 0, ^ {at+l) ^yfr {at -I) = 0, The first equation shews that t/t must always = — 0. Thus one of the undetermined functions is already elimi- nated (the other will be eliminated from consideration of the initial circumstances). Substituting in the second equation, ^ {at +1) -(j) {at -I) =: 0. Now at may, with changes of time, receive any value 48 ON PARTIAL DIFFERENTIAL EQUATIONS. "whatever. And if at—l = q (^liich makes at -\-l = q +2/); q may have any value whatever. Aud thus, q having any value whatever, we always have the equation {q+2l) =<^(?); that is, the form of the function (f> must be such that, on increasing or diminishing the quantity under the bracket by 21, the same value of the function will be reproduced. Now in either term of the expression for the displacement of the particle at x, namely (6) is a finite periodical function, going through its changes and returning to the same value when 6 is changed by 2L A trigonometric function of sines and cosines of aii angle goes through all its changes and returns to the same value when the angle is increased by 27r. Hence it appears that (j) (6) is similar in its general character to a trigonometric function of sines and cosines of — ^ — , or j 6, Any finite periodical magnitude may be represented with any assigned degree of approximation by a series of integral powers and combinations of the sine and cosine of — ^ ; and these powers may be converted into sines and cosines of multiple arcs. Thus the function cj) {6) may be represented by A^ sin ^ $ -\-A^ sin -j 6 +&c. + ^„sin -^ + &c. +B^cos-j e +B,co>i -^e + &c. +B,cos~e + &c.; Li C which may be conveniently expressed by 2(^„sin^'^] + SfAcos^^ D, E. 50 ON PARTIAL DIFFERENTIAL EQUATIONS. and thus the value of z will be 2 j J„ sm —^ '-\ +2 \ A cos — ^—^ -^j - 2 I ^„ sm ^^- ^1 -2 j 5„ cos ^-^ -^1 , or z ( z^^ COS —j— sm -^ a; 1 dz and the value of ,, will be at — Z ( zB^ sm —J- sin -^- a? ) ; -2 f^mra . . nmat . titt \ 60. Suppose now that the initial circumstances of displacement (Z) and velocity {Z') are given for every point of the string, and that from these we desire to deter- mine all the subsequent motions. We must now make ^ = 0, and we have FUNCTIONS IN PROBLEM OF MUSICAL STRINGS. 51 And our immediate object is, knowing the values of Z and Z' for every value of x, to determine the values of A^^ A^, &c., B^, B^, &c. This will be done by means of the following general formula. If we multiply sin -j- x by sin —j- x, where m is any integer different from % and integrate from a? = to a? = i, we have for product, 1 (m —n) TT 1 (m +n) ir ^ cos ^ — ~ — a;— ^cos-^^ ' x\ for general integral, I . (m —n) IT I . im -{-n) tt sm- 7—^ — X —;^, r — sm ^ ~ — x\ 2 (m —n) TT I 2 (m +7i) tt and for definite integral, 0. But if m = 71, we have for product 1 1 2/27r for general integral X I . 2n7r o - A — sin - ,- X, 2 4/i7r I and for definite integral, ^ • 52 ox PARTIAL DIFFERENTIAL EQUATIONS. Hence, if we put S for the integral with respect to x taken from x = to x = I, S(z&m^ x] = A, l\ S (Zsin—j xj = A^l, &c. S (Z' sin -J- xj = —Iran B^ ; SiZ' sin —J- X 1 = —Iran B^, &c. 61. In a given instance, in which the initial displace- ments and velocities are given not by formula but by numerical values, the following process will apply. Sup- pose we limit the multiples of y 6, where sines and cosines are to be used, to the first six, namely, TT^ 27r^ Stt^ 47r^ Btt ^ Sir ^ which will suffice in almost any conceivable case. Divide the length I into 120 equal parts, and suppose that Z and Z' are known numerically for the middle of each of these FUNCTIONS IN PROBLEM OF MUSICAL STRINGS. 5S niT parts. For the first of these middles, -j x = 45' ; for the second it is 3 x 45' ; for the third it is 5 x 45' ; and so on. And for each of these parts, the integral is, in fact, taken through the extent ^-^ . Hence, the formula above gives the following numerical rule : — Multiply the successive values of Zhy sin 45', sin 3 x 45', sin 5x45', &c. : the sum of products = 120 A^, Multiply the successive values of Z by sin 2x45', sin 6x45', sin 10x45', &c.: the sum of products = 120^2- Multiply the successive values of Z by sin 3x45', sin 9x45', sin 15x45', &c.: the sum of products = 120 J3. And so to A^, And a similar process (with proper changes) determines B^, B^, &c., from the given values of Z\ Then substituting these in the formute at the end of Article 59, the circumstances of all subsequent motion are completely obtained. 62. If A^, or B^, or both, are found to have real values, and ^.^, A^^ &c., B^, B^, &c., have no real values, then the vibration is that of a simple line of sines, de- 54 ON PARTIAL DIFFERENTIAL EQUATIONS. pending, for time, on the angle — r- . If only A^ and B^, one or both, have real values, then the vibration is that of two lines of sines ; separated at the point where -j-x ^ tt, or where x = ^ , thus forming a permanent node in the middle of the length ; and depending for time on the angle — J — , and thus vibrating in half the time of the former V vibration. Similarly if A^ and B^ only have real values, the string is divided into three equal parts by two nodes, and the vibrations are made in one-third of the time of the fundamental vibrations. The connexion, of these results, with the results of the experience of the senses as to the musical tones produced, is one of the most important points in the Acoustical Theory of Music. 63. The two instances which we have given will probably suffice, better than any rules, to shew the use that is to be made of the undetermined functions in the solution of a Partial Differential Equation. For important applications, in the Theories of the simple Echo and of Resonance, see Sound, Articles 41 and ^1. NECESSARY NUMBER OF UNDETERMINED FUNCTIONS. 00 CONSIDERATIONS ON THE NECESSITY OF CONNEXION BETWEEN THE NUMBER OF UNDETERMINED CONSTANTS AND THE ORDER OF THE EQUATION (iN SIMPLE DIF- FERENTIAL equations), and between the NUMBER OF UNDETERMINED FUNCTIONS AND THE ORDER OF THE EQUATION (iN PARTIAL DIFFERENTIAL EQUA- TIONS). 64. In solving a simple differential equation of the first order, we usually arrive at a solution containing one undetermined constant. But it is not always so. There is a remarkable class of solutions called " particular solutions," which not only have not, but which cannot have, an un- determined constant. Thus, if we express in algebraical language the problem " To find the equation to the curve whose tangents possess this property, that a perpendicular from a given point upon the tangent has a given value ;" we have, for general solution, a straight line with one undetermined constant; and for particular solution, a certain circle, which from the nature of the case does not admit an undetermined constant. The connexion, there- fore, between the order of the equation and the number of undetermined constants in its solution is not invariable. 65. But if we assume a formula as solution to a dif- ferential equation which we propose to form, the said formula containing symbolical constants, we can pre- determine that any one of those constants (if the equation be of the first order), or any two of those constants (if the 56 ON PARTIAL DIFFERENTIAL EQUATIONS. equation be of the second order), shall be eliminated from the differential equation to be formed, and shall therefore be undetermined when that equation is solved. Thus, to take a very simple instance, let y =■ ax-\-h be assumed to be the solution of a differential equation, with the constant a undetermined. Differentiating it, -,- = a ; substituting this in the given equation, y ^x -^ —h = 0, the dif- ferential equation required. This process evidently can be extended to equations of any order. 66. It appears thus that the existence of a soluble differential equation does not necessarily imply an unde- termined constant in the solution ; but the existence or assumption of an undetermined constant in a solution can always be represented by proper form of a differential equation : and that this theorem applies to equations of every order. 67. If now we apply similar considerations to Partial Differential Equations, we are led to the following con- clusions. 68. First, on examining the solution in Article 18, &c., it will be seen that, in the cases in which the solution of a Partial Differential Equation is really effected, it has been done by reducing it to an ordinary integration, just as would be done for a Simple Differential Equation. This NECESSARY NUMBER OF UNDETERMINED FUNCTIONS. 57 seems to indicate a probability that there may be cases in which the solution of a Partial Differential Equation has limitations analogous to those of a Simple Differential Equation, and therefore has not, and cannot have, an undetermined function in its solution. Upon this, how- ever, there is not at present any distinct evidence, and the matter is indicated as one which appears to merit examination. 69. Second, if we assume a formula containing one function (j>(w), where w is £i definite function of x and y and where cj) is intended to be undetermined, we can always produce a Partial Differential Equation of the first order, of which the assumed formula will be the solution. For we have 2?, which contains ^(-2^); and we can form dz . . -^ , which will contain ^{w) and (l>\w) ; -T- , which will contain ^(w) and \w) ; and from these three equations we can eliminate (l>{w) and dz (f>'{w)y leaving one equation between ^, x, y, -j- , and -jT- , of the form required. D. E. F 58 ON PARTIAL DIFFERENTIAL EQUATIONS. 70. But, third, if we assume a formula containing two functions (l>{w) and '^{s), where w and s are definite func- tions of X and y, and where (fy and >|r are intended to be xxndetermined, we cannot always produce a Partial Dif- ferential Equation of the second order, of which the as- sumed formula will be the solution. For, we have z, which contains {w) and yjr{s) ; and we can form dz -J-, which will contain <^(^^), ^'(^y), ir{s), ^fr\s) ; dz -J- , which will contain ^{w)^ ^\w), '\Jr{s), y}r\s) ; -v-2^ which will contain \w), (f>''{w)yyjr{s), dxdy d^ , which will contain the same six functions ; which will contain the same six functions. Here we have six equations, and no more, with six functions on the other side of the equations. We cannot therefore eliminate these six functions, so as to leave an tequation between d2 db ^ d^z d^ ""'^'^'dx' dy' dx^' dxdy' dy'' NECESSARY NUMBER OF UNDETERMINED FUNCTIONS. 59 And thus we cannot always produce the Partial Dif- ferential Equation required. The same remark, it will be found, applies more strongly to equations of a higher order. 71. It seems not impossible that the failure of at- tempts to solve the equations in Article 44 may have some connexion with this inability to establish a perfect con- nexion between the order of the equation and the number of undetermined functions. 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