378 
OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
These operations correspond to the rejection of terms involving i 3 . 
We thus obtain, in lieu of equations (29) to (34), which involve the unknown true 
coefficients on both sides, others of the form 
ih=Pi+A j+Aj?, 
£ = Qi + IM + B t i 2 , 
#= P 2 + A 2 i -f- A 2 i 2 , 
5'2=Q 2 +B^-l--5 2 i 2 , 
£3 = P3 + A 3 ? -f- AA 
£ = Q3 + + B 3 i 2 , j 
&c.=&c., 
(36) 
in which A 1? A 19 B 1? B l9 &c. are numerical quantities. 
The true period (and therefore i) being known, these expressions give the values of 
the coefficients for the true period in terms of those for the approximate period ; and 
these values being inserted in equation '(13), it will then express the phenomenon for 
the true period in terms of the coefficients for the approximate period. The general 
expression for A, and A Y &c. would be too lengthy to write in full, although the calcu- 
lation of their numerical values in any particular case is not very tedious ; the most 
convenient mode of procedure is to work out, by equation (35), the numerical values of 
the second approximations to#, £, &c., and insert these in equations (29) to (34). 
14. To illustrate the application of the method described, and to show that advantage 
is gained by it, we have chosen, arbitrarily, the law of periodical variation 
or 
■where 
a m ~ — cos(?w;3+60 0 )+cos2mz— cos 3 mz 9 
a m = — - 5 cos *86603 sin mz -\- cos 2mz — cos 3 mz, 
# = -•50000; £=+*86603; #,=+1-00000; £=-00000; #= + 1-00000; 
£=• 00000 ; 
and taking z=30, i— 5', and ^=120, we have calculated one hundred and twenty suc- 
cessive values of a m , corresponding to the successive values of 2—0° O', 30° 5', 60° 10', 
&c. .... (3570°+ 9° 55') ; then, treating these numbers as if they corresponded to values 
of z — 0°, 30°, 60°, &c 3570°, and applying to them Bessel’s method, the following 
values of the approximate coefficients were obtained : — 
P,= --42262; <+=+-90398; P 2 = + -97128, 
Q 2 = — "17383 ; P 3 = — -96147 ; <+= + -25536. 
With these values, and the other data which supplied them, equations (35) and (36) 
give as third approximations to the true values of the coefficients, 
#=-•50000; ^=+-86604; #=+1-00011; 
£= + •00002; #=--99995; £= + -00002; 
