On a Method of Tracing Periodicities. 



3G1 



" On a Method of Tracing Periodicities in a Series of Observa- 

 tions when the Periods are unknown." By VlNAYEK 

 Narayeu Nene, First Assistant, Government Observatory, 

 Colaba, Bombay. Communicated by C. Chambers, F.R.S., 

 Superintendent of the Observatory. Received May 30. 

 Read June 15, 1882. 



[PlATE 4.] 



1. The object of this paper is to investigate a new method which will 

 enable us to find out the period of an unknown inequality, and thence 

 the inequality itself, from a series of observations — such as magne- 

 tical, meteorological, or astronomical — when they are of a periodic 

 nature. 



2. We assume that there is some large complex period, such that 

 the observations — which need not necessarily correspond to equidistant 

 times, but must have the time similarly divided in each general period — 

 will be exactly repeated after such full period ;* but since we have not 

 yet found such a period, although we have observations of thirty years' 

 duration or more, say, of barometer, we may say that such a period is 

 not yet completed, and we cannot, therefore, apply Bessel's method, 

 which is applicable to only known periods, to find subordinate 

 periodic inequalities. Diurnal and annual periods which have already 

 been discovered may be considered as subordinate periods of the large 

 period which we are considering. Our aim is first to ascertain the 

 duration of subordinate periods one by one, and then to find out the 

 amplitude for each period. 



3. Any observations of a periodic nature may be represented by 

 the equation — 



T=^+^ 1 sin(u i + ?^ + ^sm(^U 2 +^+, &c. . . (1), 



in which K 1? K 2 , &c, which are all unknown periods which are in an 

 ascending order of magnitude, but the relations of which with each 

 other are also unknown ; Y is the phenomenon, and x the variable 

 time on which the phenomenon depends ; u, Ui, u 2 , and Ui, TJ 2 , &c, 

 are constants ; and 27r is of course equivalent to 360°. 



4. The form given in Bessel's paper, embodied in the " Quarterly 

 Weather Report of the Meteorological Department," Part IV, October 

 to December, 1870, is — 



Y=u+ Ul sin (iTi + ^ + u 2 sin (l\ + ^ + , &c, 



which form can also be written — 



* Practically we shall deal only with observations that subdivide the large period 

 spoken of into equal intervals. 



