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XIV. On the Mathematical Expression of Observations of Complex Periodical Pheno- 
mena; and on Planetary Influence on the Earth's Magnetism. By Charles 
Chambers, F.B.S . , and F. Chambers. 
Received May 26, — Read June 19, 1873 *. 
The writers purpose in the following pages to determine, by Bessel’s method, a mathe- 
matical expression for a periodical phenomenon from observations which are affected 
by one or more other periodical phenomena, and to find criteria for judging of the 
extent to which the expression is affected by these other phenomena ; also, having 
found an expression for a period of known approximation to the truth, to find from it 
the expression for the true period. In the course of these inquiries, certain ambiguities 
which affect similarly Bessel’s expression for a single periodical phenomenon and the 
results here arrived at will be remarked upon ; and, finally, the results will be applied 
to determine the nature of periodic planetary magnetic influence in particular cases. 
2. In Bessel’s paper “ On the Determination of the Law of a Periodic Pheno- 
menon ” (a translation of which has been published by the Meteorological Committee 
in the Quarterly Weather Report, part iv. 1870), the author describes, in Section VII., 
how periodical phenomena which depend on two or more angles can be developed 
from observations of the same ; and he remarks upon the simplicity of a certain class 
of cases in which both angles are exact measures of 2sr, and one is a multiple of the 
other. In the description of the process occur the following words : — 
“ If we designate the two angles by x, x 1 , then in the expression 
yz=p-\-p l cos^H-gi sin #+p 2 cos2#-{-g , 2 sin2,r-|- See. 
the p, p x , q x , See. which occur are not constant, but depend on x 1 ; and as they are 
periodic functions of x ', each of them has an expression of the form 
a-\-a x cosaf+^i sm.od-\-a 2 cos 2#' -j -b 2 sin 2x’-\- Sec. 
It is therefore necessary to deduce this development of p, p x , q , &c. from the obser- 
vations. If the available series of observations gives the values of y, not only for 
values of x (0, 2 , 2z, ... . (n—l)z), which are in arithmetical progression and fill 
up the period, but also for the combination of each of these values of x with n- values 
of 4/(0, z', 2z\ ( n ' — l)s f ), fulfilling the same conditions, the development has 
no difficulties.” After a perfect elucidation of a type of these cases follow remarks 
upon comparatively difficult cases, which require more cumbrous methods for elimi- 
nating the several constants. 
MDCCCLXXV. 
Subsequently revised by the authors. 
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