the whole subject it is convenient to deviate somewhat from the historical order of development, and to begin with the consideration of impulsive motion. Whatever the actual motion of the system at any instant, we may conceive it to be generated instantaneously from rest by the application of proper impulses. On this view we have, if x, y, z be the rectangular co-ordinates of any particle m,
mx` = X', my` = Y', mz` = Z', (2)
where X', Y', Z' are the components of the impulse on m. Now let [delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y, z which are consistent with the connexions of the system, and let us form the equation
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = [Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z), (3)
where the sign [Sigma] indicates (as throughout this article) a summation extending over all the particles of the system. To transform (3) into an equation involving the variations [delta]q1, [delta]q
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