DEPARTMENT OF TERRESTRIAL MAGNETISM. 319 



DETAILS OF WORK IN WASHINGTON. 

 INVESTIGATIONAL AND PUBLICATION WORK. 



On pages 301-303 a general account is given of the various researches 

 under this head. A fuller statement respecting the chief investigations 

 is contained in the following abstracts : 



On the study of methods of seeking hidden periodicities, with some applications to magnetic 

 and Sun-spot data.^ G. N. Armstrong and C. R. Duvall. 



The main part of the work was devoted to recurring-series methods, which 

 were first investigated by Lagrange in 1772. In his dissertation^ the first 

 of the authors developed these methods, with numerous applications (1) to 

 computed values of an assumed periodic function for the purpose of testing 

 the powers and limitations of the methods; and (2) to observational data of 

 suspected periodicity. 



Assuming that any periodicity in a series of numbers may be expressed as 

 a sum of sine terms, the number of different periods being equal to the 

 number of sine terms in the sum, it is shown in the recurring-series method 

 that the necessary condition for the existence of periodicity in the given 

 numbers is the vanishing of certain functions of these numbers. It was found 

 that the most convenient form for applying the vanishing conditions is that 

 of determinants made up of the given numbers in the manner explained in the 

 "reciprocal scale" method of the dissertation.^ If all such determinants of 

 the (71 + 1)'*' order vanish, the condition is satisfied for the existence of n dif- 

 ferent periods in the given numbers. The solution of certain linear equations, 

 greater than n in number, gives the n "scale coefficients." The method of 

 least squares may be applied in this solution. An equation of the n^^ degree 

 in one unknown is then formed, the coefficients being certain simple functions 

 of the "scale coefficients." To each real root of this equation, which is not 

 greater than 2 in absolute value, there corresponds an independent period of the 

 given numbers, the cosine of the period angle being equal to one-half of the 

 root in question. Other values of the roots correspond to exponential and 

 hyperbolic functions. 



The great difficulty in the application of the method is encountered when 

 the vanishing of the determinants is tested. Even for computed values of a 

 function of known periods, the failure to vanish may be surprisingly large. 

 Here the only errors in the given numbers are due to the "rounding off" 

 process, and the failure to vanish is entirely due to accumulations from these 

 errors. In the application to observational material the trouble with the van- 

 ishing of the determinants is likely to be very much greater. Not only are the 

 given observations uncertain, often by unknown amounts, but there is gener- 

 ally also a question of the validity of the assumption that a periodic function 

 of a specified form may be made to represent these observations. It is 

 extremely desirable, therefore, to have some criterion by which to judge of 

 the exactness with which the determinants should vanish. 



About the time the present work was undertaken, an article came to hand in 

 the IMay (1914) number of the Monthly Notices of the Royal Astronomical 

 Society, by Professor J. B. Dale, in which just such a criterion was proposed. 



'Abstract of work done during July and August 1914, at the Department of Terrestrial Mag- 

 netism. Fuller publication is to be made in the journal Terrestrial Magnetism and Atmospheric 

 Electricity. 



^Armstrong, G. N. Eine Untersuchung der Anwendbarkeit rekurrenter Reihen zur Aufsuchung 

 versteckter Periodizitaten. Miinchen, Dissertation, K. Technische Hochschule, 1913 (pp. 96 with 

 figs.). 24 cm. 



