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.

128 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
V. CALCULATION OF AMPLITUDES IN SPECIAL CASES.
The Fourier coefficients having been calculated for the 26 and 27 day periods in each
year, we are able to obtain the amplitudes for periods not differing too much from these
values. To shew the process of calculation to be adopted for this purpose, let A1, A2,
etc.; B1, B2, etc. be defined by the equations
0n T 22nT snT
A1 = jf (t) cos gtdt, A = f(t) cos gtdt, Am=J f(t) cos gtdt,
^O ' nT ( s-l) nT
çnT -2nT fsnT
B, = f(t) sin gtdt, B2= f(t) sin gtdt, B = f (t) sin gtdt,
= JJnT (s- 1) nT
where g = 27r/T.
It is required to find
2 P T' 2 PT
a' = j (t) cos Ktdt, ' = (t) sin tdt,
where Kc = 2Tr/T'.
If K and g do not differ much from each other we may put approximately
înnT mnT
f (t) co t ft) os (gt + ) =, cos a,, - B sin a,,...............(4).
(m-1) nT J (m-1) n
The greatest approach to equality is assured when the curves cos Kt and cos (gt + ca)
are made to coincide as nearly as possible throughout the interval, and hence the phases
should agree in the middle of the interval, so that for t= (m - )nT, Kt=gt+.
This gives
a = 27r - (-) nT.
We may now put
"nT îmn=s m=s
f (t) cos Ktdt = 4 A, cos amr - S B, sin a,,,
0 mb=l m=l
l'snT. m=s m=s
f(t) sin Kctdt = A,,, sin.,, + 1 B, cos ar.
-m0 il=1 m=l
The coefficients which we suppose to have been calculated are
a, = A, b= - B,, etc.,
2 f sIT 1
so that s nT J /(t) cos,ctdt = - S (a,, cos a,, - b,, sin an).?îZ o l s