dF the second multiplied by and sum the result with respect dpi to i from i=1 to i=n inclusive. We have dF' Φ dF dΦ do dF dø dpi dF do dp;) (8). dp; dpi dxi dpi dp; dx;/ The expression under the double sign of summation in the second member vanishes when i=j; we may therefore restrict the summation to unequal values of i and ;. Now as for any particular combination of values, e.g. 2, 3, there would exist in the completed member both the terms corresponding to i=2, j = 3, and those corresponding to j=2, i=3, it is evident that if we employ the symbol is to denote summation with respect to different combinations of i and j, the second member of the last equation may be expressed in the form dF do dp;_dF do dp; dФ dp, dp: dxdp: dp; dx. ; ; dpi dpi dp; dpi) (dx; dx; so that the equation (8) becomes dF' Φ dF (Φ) 9. \dpi dpi dpi ij + The number of terms of which the second member ex n (n − 1) presses the sum is thus and it will be observed that 2 > as to any particular term it makes no difference in what order the numerical values of i and j are assigned to these quantities; e.g. whether for the combination 2, 3 we make i= : 2, j=3, or i= 3, j=2; but we must confine ourselves to one order. Now when the equation dz=p, dæ, +p, dx, +... dx+ +Pndum is integrable in the manner here supposed, we have for all combinations of i and j, dpi _dp dx; dx; All the terms in the second member of (9) therefore vanish, and we have (dF do dF do ' dΦ Σ. = 0. dx; dpi dp: dx;/ This is the direct form of the Proposition under the particular limitation supposed. As F, Q represent, under the same limitation, any two of the first members of the n equations (3), (4), which determine n (n-1) Pu, ... Pa, there will exist equations like the above. It is usual to employ for brevity the notation dF 4Φ dF dΦ do =[Fo], and this being done the above system of equations expresses the n (n − 1) functions of the form [FF,) as linear homogeneous ;] 2 n (n 1) functions of the dpi_dpi quantities of the form dx; dx;' It is hence that the vanishing of the latter series of quantities secures the vanishing of the former. The converse truth will therefore be established by shewing that the n(n-1) dpi_dpi quantities of the form dx; dxi 2 2 are, when 2 n 2 2' 3 1 dF dF, 2 3 3 F, F., ... F, are independent with respect to Pu, P2, ... Png F2 n (n − 1) expressible as linear homogeneous functions of the functions [F;F;]. To avoid complexity of expression I shall establish this for the particular case of n=3, and shall shew that the reasoning is general. The functions Fi, F,, Fy, being independent with respect to Pu, P2, P3, the determinant dF dF, di dF. dpi dpa dp does not vanish. This determinant we shall denote by A. In (9) writing for Fand o first F, and Fg, secondly F, and Fı, thirdly F, and Fy, we have on changing signs the system (, , _ , – ) +.. dpa + dp, dp, dp, dp,/ dx, dx, , , \, , , /dp - [FF]=( t... (10). \dp, dps dp, dp, ldx, / 3.72 , \/ dF, Multiply the first equation by 1, the second by dF, third by 3 and add. Then dp, _dp.) \dx, dx, 2 dps 3 -[FF]=(1) 3 2 3 dps dpi as a whence as A does not vanish we have, on dividing by it, the dp, dp, function dx, dx, expressed as a linear homogeneous function of [FF], [F;F], and [F,F.]. dF dF, DF, In like manner multiplying the equations by dp, dp,” dp, ', respectively, and dividing by A, we obtain ap: – dx, dr similar linear homogeneous function, and lastly, multiplying dF dF, dF by and proceeding as before, we obtain dpz' dps' dpz dx, dx, as a similar linear homogeneous function. From all which it follows that when [F,F], [FF], [FF] vanish, then dp, dp, dp, dp. da, 2 dp, dp, dp, dp, dx,” will vanish also. 2 The reasoning is general in its nature. If Fi, F,, ... Fn are independent with regard to PP, ... Pn, the determinant dF, dF, (1), dF dF ...... n as does not vanish. This determinant is from its constitution a determinant linear and homogeneous, not only with respect to any row or column of elements, but also with respect to the possible binary combinations which can be formed of two rows or columns, ternary out of three rows or columns, &c. provided that these combinations are themselves of the form of determinants. In the language of the theory such combinations are called minor determinants. Hence if we construct the system of equations represented by (10), and observe that the coefficients of any particular term of the form dpi _dp, in the several equations form a system of such ; dx; dx; minors to the general determinant (11), it will be plain that the equations can by multiplication and addition be brought to a form in which the coefficient of that particular term will be A. At the same time the coefficients of all the other terms of the form dpi _dp; will vanish. For a little atten dx; də; tion will shew that they will be what the determinant A would become on making two of its columns or rows of elements equal, and therefore will be identically equal to 0. Thus the Proposition is generally established for the case in which z does not explicitly appear in the functions F,, F, ..... F, When z does appear in those functions the equations (6) will be replaced by dF dF dF dp, dF dp = 0, + + dxi dpn dx; + Pidz+ dp, dxi do dpı + ... + dФ dp. n + Pi da do do , do dp, 0, dpn dxi from which it is seen that the theorem above established will only need to be changed into the form employed in the statement of the general Proposition. As the above is one of the most important propositions in the entire theory of Differential Equations, it may be desireable to illustrate it by examples. [There are no examples in the manuscript.] 4. We resume the general theory. The integration of non-linear partial differential equations may be effected by two distinct methods, both resting upon |