43]
WHICH OCCUR IN DYNAMICAL PROBLEMS.
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Suppose that u and v continue to represent arbitrary functions of x, y, z,... but that the
remaining functions w, ... are such as to satisfy W — 0, ... (so that w,... may be
considered as the constants introduced by obtaining all the integrals but one of the
system of differential equations in x, y, z, ...), we have
dMVU dMVV
du dv
V
,d.MX dMY dMZ
V dx dy dz
Also the only one of the transformed equations which remains to be integrated is
du : dv = U : V, or Vdu — Udv = 0,
(in which it is supposed that U and V are expressed by means of the other integrals in
terms of u and v).
Suppose M can be so determined that
dMX dMY dMZ
dx dy dz ’
(M is what Jacobi terms the multiplier of the proposed system of differential equations):
then
dMVU dMVV
du + dv
or MV is the multiplier of Vdu — Udv = 0, so that
j MV (Vdu — Udv) = const.
Hence the theorem:—“ Given a multiplier of the system of equations
dx : dy : dz, ... = X : Y : Z ...
(the meaning of the term being defined as above), then if all the integrals but one of
this system are known, the remaining integral depends upon a quadrature.”
Jacobi proceeds to discuss a variety of different systems of equations in which it
is possible to determine the multiplier M. Among the most important of these may
be considered the system corresponding to the general problem of Dynamics, which
may be discussed under three different forms.
§ 4. Lagrange’s first form 1 .
Let the whole series of coordinates, each of them multiplied by the square root
of the corresponding mass, be represented by x, y, ... and in the same way the whole
series of forces, each of them multiplied by the square root of the corresponding mass,
by P, Q, ... ; then the equations of motion are
drx _ Y d?y __ y
dt* ’ dt>~
1 I have slightly modified the form so as to avoid the introduction of the masses, and to allow x (for
instance) to stand for any one of the coordinates of any of the points, instead of standing for a coordinate
parallel to a particular axis.
