OP OBSERVATIONS OE COMPLEX PERIODICAL PHENOMENA. 
369 
8. If instead of applying Bessel’s process at once to each individual obser- 
vation, we had begun by finding a mean value 2 ^ (as affected by the other pheno- 
menon) of the one phenomenon at a particular phase of its period x, and then proceeded 
to apply Bessel’s process to the^ mean values of this character, we shoulcl,have arrived 
at precisely the same results. 
We might also have regarded a hypothetical complex phenomenon of period gx as 
being produced solely by the recurrence of the phenomena whose periods are x and x', 
and finding by Bessel’s process from the r observations the coefficients of its expression 
— from these determining the coefficients of the expressions for the component periodical 
phenomena; this, too, would have led to the same results. 
9. To conclude this section, we draw from what has preceded the following practical 
rule for deducing from a series of observations of the combined effect of several inde- 
pendent phenomena (observations taken at equal intervals of time) the coefficients of 
Bessel’s series for each separate phenomenon : — Find the least integral numbers f, g, h, 
See. which are proportional (or nearly so) to the periods x, x\ x", &c. of the several 
phenomena, and let v be the least common multiple of those numbers ; choose then for 
treatment observations extending exactly over some multiple of the period^, and note 
whether any values of jp s or q s , P s or Q s , &c., for which s is small, other than the first 
terms, enter into the equations (7) ; if not, proceed to apply Bessel’s method to deter- 
mine from the observations the coefficients of the expression of each phenomenon, just 
as would be done if the observations were unaffected by the other phenomena. 
II. 
10. It will be useful further to estimate in what degree the phenomenon whose 
period is x' affects the values of the constants ^> 15 q t , &c., in the expression of the phe- 
nomenon whose period is x, when the number (K) of observations is greater than and 
not a multiple of r. And here, confining our attention to strictly and exclusively 
periodical phenomena, we must reject the constant term (^o+Po) in the expression for 
the combined phenomena: this is equivalent to substituting for the original obser- 
vations a 0 , « 15 a 2 , a m the excesses of them respectively above their mean value 
2 p-. Let — =E= = ^-, c being an integer, and let cx=(d-\-e]f'ic', d being integral and 
e a proper fraction. If we represent (3 m by the general term 
\_p 5 cos smz-\-q s sin smz\, 
and y m by the general term 
£p* cos s ^ mz + Q s sin s - mz \ , 
3d 
MDCCCLXXV. 
