105]
HARMONIC SCREWS.
97
Suppose the body to be in motion under the influence of the forces, and that
at any epoch t the co-ordinates of the twisting motion are
d0j d0n'
di ’ dt ’
when referred to the principal screws of inertia. Let £/', ... be the
co-ordinates of a wrench which, had it acted upon the body at rest for the
small time e, would have communicated to the body a twisting motion
identical with that which the body actually has at the epoch t. The
co-ordinates of the impulsive wrench which would, in the time e, have pro-
duced from rest the motion which the body actually has at the epoch t + e,
are:
t" + e f " 4- p
+e dt + e TlF'
On the other hand, the motion at the epoch t + e may be considered to
anse from the influence of the wrench ... for the time e, followed by
the influence of the evoked wrench for the time e. The final effect of the
two wrenches must, by the second law of motion, be the same as if they
acted simultaneously for the time e upon the body initially at rest.
The co-ordinates of the evoked wrench being :
1 dV 1 dV
2?i do.+2Plld0,r
we therefore have the equation :—
or
1 dV
2pi d0j ’
= 1 IF
dt + 2p, døj ’
■ and n — 1 similar equations; but we see from § 97 that
pj dt ’
Differentiating this equation with respect to the time, and regarding e as
constant, we have
whence
e = ■
dt pj dt2
2Muj2<i^ =
dt2 d0j
the same equation as that already found by Lagrange’s method.