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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
2

do
Σ
dx; dpi dpi dxi

dF dpi dF do dp;)
ΣΑΣ,

(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;
d


Σ,

dp, dp: dxdp: dp; dx. ;
dF do dp: _ dF dpi)
dpi dp; dx, dp; dpi dx;)

;
dF dF dø\/dp: _ dpi

dpi dpi dp; dpi) (dx; dx; so that the equation (8) becomes

dF' Φ dF (Φ)
Σ.
dx; dpi dpi dx;
dF do
dF \ dpi dpi

9. \dpi dpi dpi

ij

+

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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

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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, , +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

' 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 dF

do
Σ
lor dpi dpi dx;).

=[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

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n

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2

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2'

3

1

dF

dF,

2

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3

3

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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
dp,' dp.' dp

dF.
dpi' dp dp:
dF dF 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

(, , _ , ) +..
dFDF dF, DF

dpa

+ dp, dp, dp, dp,/ dx, dx,

, , \,

,
dF, JF, _ dF dF) du

, /dp - [FF]=(

t...

(10). \dp, dps dp, dp, ldx,

/

3.72
dF, DF,_ dF dF, dp, _dp

, \/
– [FF]=(ap, dp, - dp, dp:/\dx, dx,
dp
dF

dF, Multiply the first equation by 1, the second by

dF, third by

3 and add. Then
dF
dF.

dp,

_dp.)
[] FF
dp.
dpi

\dx, dx,

2

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dps

3

-[FF]=(1)

3

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2

3

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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.
dx, dx, dx, dx,

da,

2

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dp,

dp,

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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,
dp, dpm
- A

(1), dF

dF

......

n

[blocks in formation]

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; ; 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.

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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 + ... +

dp.

n

+ Pi da

do

do , do dp,
+

0,
dx;
dp, dx;

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

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