n

This is wholly independent of the relations connecting u, u,,...u with x, x,,..... Now choose the n-2 variables un- so that u1 = c1, u2 shall be integrals of the given partial differential equation (a). Then that equation transformed becomes

29...

19

=

C2

• Un-2 = Cn-2

of which the auxiliary ordinary equation is

Undun Un-du„ = 0.

At the same time the equation (6) becomes

_d_(MU) + d (MU.) = 0.
(표)

dun-s

H n-1 du

Hence is the integrating factor of the preceding diffe

M
H

rential equation between un-1

and un

Jacobi's theorem in its most general form is thus seen to be the following

be transformed by the introduction of a new system of variables u,, u,,....un, so chosen that

shall be integrals of the given system, then the final differential equation between u-1 and u, shall have for its integrating

M

factor in which M is any function satisfying the partial

H'

The form of Jacobi's theorem obtained by the previous demonstration may be deduced from the above by choosing for u, un two of the original variables, for example -1, 2? and transforming the integrals u,, u,,....un so that u, shall contain only x xn, us shall contain only x....x, and so on.

... 'n

Examples.

3. Jacobi has established by means of the above theorem the very remarkable theorem that in any ordinary dynamical problem the forces depending not upon the time but upon the material constitution of the system, if all the integrals but two of the dynamical equations are found, the two remaining integrals can be found by quadratures.

1st. In a dynamical system of free points the forces acting upon which depend only upon the position of the points, we have if we represent the entire system of rectangular coordinates taken in any order by x, y, z,... and the corresponding resolved forces divided each by the corresponding mass by X, Y, Z,... the system of equations

Now as X, Y... do not contain t we may consider first the system

and it is evident that if we can find all the integrals of this system, t will be given by the equation

having been first converted by means of the supposed integrals into a function of x.

To determine the last multiplier of the system last written we have first the equation

which since X, Y... do not contain x', y'... is satisfied by Ma constant. Giving to the constant the particular value 1, we see that if

are n-2 integrals of the system, and if by means of these we eliminate n 2 of the variables and construct the differential equation between the two remaining variables, the integrating factor of that equation will be in which H is the functional determinant of u,, uz, .... Uμ•

Un

1

H'

2ndly. Suppose the system subject to a material connexion which establishes an equation of condition among some or all of the co-ordinates. If we represent the co-ordinates taken in any order and multiplied each by the square root of the corresponding mass by x, y,... the corresponding resolved forces by X, Y,... and the equation of condition expressed by means of the above modified co-ordinates by = 0, the differential equations will be

the transformation above employed reducing all the equations to the same type. [See the next Chapter.]

αφ

d (Me)+ d (My) d {M(x+x dd)}

dx

....

+

dy

dx

dx'

....

Now does not contain x', y'.... Let us inquire whether it is possible to determine M also as a function of x, y without x', y'.... so as to satisfy the above differential equation.

8M+ M (dy do + dy do + ....) = 0,
dx dx dy

and from this we must eliminate λ.

Now since =0, we have by differentiating and putting