Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

TRANSVERSAL VIBRATION IS MECHANICALLY POSSIBLE. 327
(omitting those in the same line, as their attractions are
equal and in opposite directions): and suppose them to be
attractive, and as the inverse square of the distance: and
the absolute force of each = m. The whole force tending
to pull A downwards is
m (h+ u - u,) m ( -u,) m (lh - u + u,)
h +(- 4 -t + le _- {h(u)hl+ (-2}i - {h+ (/t-u +1) I
m (h + u - ') m(u-u') m(h-u +u')
+ h2+ (h+u — ) + {h -U h2(h-u+u1)'
Expanding these fractions, and neglecting powers of u - u
and u - u' above the first, the force tending to diminish u is
~ 1\ m
(1 -.) ^h3(2U - u,- u).
Putting for u,
duz dc2' h
u - h + —
dx dhoe 2'
and for u',
du d':u h'
+ -- h +.,
dx dx' 2
we find
d2cu 1 m du*
dt 2-] h dX
an equation of exactly the same form as that for the transmission of sound (10). The solution therefore has the same
form: and therefore the transversal motion of particles supposed here follows the same law as the direct motion of the
particles of air: that is, it follows the law of undulation.
* If h is so large with regard to the length of a wave that the terms after
h2 cannot be safely neglected, we may, by assuming a form for the function ex.
pressing u, integrate the equation
d -( - _ (= u l-U, - h u').
Irf,