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.

DEVELOPMENT OF R. 111
(4) In the product of a power of v by a power of
v', the product of two such terms as
e". cos x(nt + e - ) and e''ceosm'(n't + e - )
will produce
e e' cos x (n t + e - Zr) + x' (n' t + e' - lr')
and e e'v' cos x (nt + e - w) - x'(n't + e' - r').
Tracing in the same way the effects of multiplying these by
another cosine we shall get this general rule: the principal
coefficient of cos v (n t + e - ar) <r' (n' t + e' - a-') is a multiple
of ez e'.
(5) Suppose now we consider how in the expansion of
R such a term as cos (l3n - 8n')t + Q is to be produced.
It might be produced by
7(at +cn-t+ '), 6(nt + e-'), and (nt + - ').
This (by what has gone before) would have for coefficient
a multiple of e e'; of the seventh order. But it might be
produced by
8 (nt + e - nt ), 5(nt + e ~- r)
9 (nt + e- 't+'), 4(nt + e - ), ('t + -')
10(nt+e -n't + e'), 3(nt+e-ar), 2(n' + e-r')
il (nt + e - n't + E'), 2 (nt + e - ar), (n't + e'-r')
12 (nt + e - n't + e'), (nt + e - r), 4 (n't + E'- ar')
13 (nt + e - z't + e'), 5 (n't + E'- r');
and the coefficients of these combinations would be multiples
respectively of e5, ee', e e', e2e, ee'4, e'5: all of the fifth
order: and 13-8=5. By trying any other combinations it
will be seen that their coefficients are of the seventh, ninth,
or some higher order, but that none can be formed with
coefficients of a lower order. And as the sane reasoning
applies in all other cases, we easily arrive at this general
conclusion, that
the principal coefficient of a term cos (pn - qn')t + Q in the
expansion of R is of the order p - q.