CHAPTER XXXI. THE JACOBIAN THEORY OF THE LAST MULTIPLIER. 1. A SYSTEM of n differential equations of the first order and degree containing n+1 variables admits of n integrals of the form 29 uu,...u being independent functions of the original variables. When n-1 of these integrals have been found they enable us to eliminate n - 1 variables, with their differentials, from the given system of equations, and so to obtain a single final differential equation of the first order between the two remaining variables. The final equation admits of being made integrable by a factor, and its solution so found would constitute the nth and last integral of the system. We propose in this Chapter to develope the theory of the above integrating factor as established by Jacobi. The term 'principle of the last multiplier,' which is more usually employed, seems objectionable; for the essence of Jacobi's discovery consisted not in demonstrating the existence or the nature of the last integrating factor, but in the peculiar form of the method which he gave for its determination, and in the relations which are implied in that form. The discovery may be briefly said to consist in this; viz. that instead of forming by means of the n-1 known integrals the final differential equation between two variables and applying methods analogous to those of Chap. V., to determine its integrating factor, we construct antecedently to all integration a linear partial dif ferential equation of the first order, any one integral of which will enable us to assign an integrating factor of the final differential equation, whatever the order of the previous integrations may have been. Again, this partial differential equation depending for its construction only upon the form of the system given, we can often by examining it affirm beforehand that if all the integrals but one of the system be in any way found, the final integral will be deducible by quadratures. This happens in the case of the most important of all systems of differential equations-that of Dynamics. Further, an ordinary differential equation of the nth order being reducible to a system of n differential equations of the first order, Jacobi's theory may here also enable us to predicate the possibility of the last integration when the previous integrations have been effected. Beginning with a single differential equation of the first order reduced to the form in which X and Y are functions of the two variables x and y, we know by Chap. v. that the integrating factor μ will be given by the solution of the partial differential equation Consider next a system of two differential equations of the first order expressed in the general form X, Y, and Z being functions of the three variables x, y, z, and suppose one integral, represented by to be known. The function (x, y, z), or, as we shall express it for brevity, 4, will obviously satisfy the partial differential equation dx + x dø dy аф ..(4), of which indeed the given equations form the Lagrangean auxiliary system; see Chap. XIV. If from the given integral we determine z as a function of x, y and c, and substitute its value in the first of the given differential equations, viz. the latter will be converted into a differential equation between x and y. But we may leave to the equation its prior form, provided that we regard X and Y as functions of the variables x and y, both explicitly as they appear therein, and implicitly as they are involved in z. And this being so, the equation (1) will become d dz + dx + + = 0. dz dy dz dz The values of and in this equation must be found dx dy from the known integral (3); they are dz аф аф dz dx dx dz' dy substituting which we have d (μX) do__d (μX) do, d (μY) do __d (μ Y) do + dx dz dz dx dy dz dz dy =0...(5). This then is the partial differential equation for determining μ. But the construction of this equation supposes to be known. We propose to shew that u can be determined by a process in which the only partial differential equation to be solved can be constructed without the knowledge of p. Now adding the last three equations together we see that the first member of the result vanishes by (5): we have thus The second line of the first member is equal to d d αφ = = 0. and therefore vanishes by (4). There remains then de (ux do) + d (uxdb) + d (uz do). dx dz dy dz dz If then by the solution of this equation a value of M dis M tinct from O be found, the function will be an integrating аф dz factor of that final differential equation which remains when z has been eliminated from the system (2) by means of any known integral = c. It will be observed that the equation for M is analogous in form to the equation for μ in the previous system. And this suggests the form of the general theorem. Thus proceeding to the case of a system of three equations will virtually involve only the variables x, y, z, since t |