Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 129
If snTT=pT' and p is an integer, the left-hand side would represent the coefficient
of period T' obtained by analysing the record of p successive periods. If p is not an
integer we may still take this to be approximately the case if sn is large, for we
may always put
snT iT' (p+e) T'
f (t) cos Ktdt f(t) cos t dt + f (t) cos Ktdt.....................(5),
0 ' O0 pTI
p being the nearest integral to sn, and e a fraction. The second integral will be small
compared to the first, if the first includes a large number of periods.
We have therefore finally for the required coefficients a' and b'
a' -m-, (a, cos a m-bm, sin ma=,rmoS(an + m),
pT m=1 pT
(............(6),
nT m^s nT
b' = -, 2 (a, sin a,, + bm cos aqr) =, rm sin (cc. + Cm),
pT' =1
where a, = r, cos m, bm = rm sin qm.
The fraction nT/pT' may generally be taken to be equal to 1/s.
The values of a are those given above, so that
a = 7rn - -, 2 -= 37r T', a= (2m - 1) rn T...............(7).
It remains to be shewn what error has been introduced by the assumed equality
(4) and the neglect of the second integral of (5). For this purpose we imagine the
function f(t) to be accurately represented by cos/ct, so that
mnT 2Ko KT
Am =, cos ~cos t os gtdt = - sin -- cos K (rn - ) nT,
(m-1)nT K9 -
and as a = ( - g) (m - ) nT,
Al= ~+ g sin IcnT cos an,
where the lower sign is taken when n is odd.
Similarly
2g7
Bm = + _ 2 sin -~KiT sin am,.
K 2 -2 2-g
By substitution it follows that, using equations (6),
2 2
a'= T' 2 _2 sin i-nT (K/ cos2 au, + g sin2 qaQ)
2 1
= T 2 2 sin ~ /crT ~ {(K + g) + ( -/Y) cos 2an},
VOL. XVIII. 17