XIV. On the mathematical expression of observations of complex periodical phenomena; and on planetary influence on the earth's magnetism
The writers purpose in the following pages to determine, by Bessel’s method, a mathematical 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 Phenomenon” (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 2π, 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’ , then in the expression y = p + p 1 cos x + q 1 sin x + p 2 cos2 x + q 2 Sin2 x + &c. the p , p 1 , q 1 , &c. which occur are not constant, but depend on x'; and as they are periodic functions of x', each of them has an expression of the form a+ a 1 cos x' + b 1 sin x '+ a 2 cos2 x' + b 2 sin2 x' +&c.