362 



Mr. V. N. Nene. On a Method of Tracing 



Y=u + iv l sin 



(u 1 + ^) + « 3 sin(u 2 +^ + , &c. . 



(2), 



2 



in which K represents a period, after the completion of which the 

 phenomenon Y constantly recnrs, and x is the variable quantity on 

 which the phenomenon dejDends. The given condition is that Y 

 remains constant for valnes of x, which are increased or diminished 

 by K, 2K, 3K, &c, u, u 2 , and U 1} TJ 2 , &c, being constants, and 27r 

 equivalent to 360°. 



5. Comparing the form of equation (1) and (2) it will be seen 

 that they are both alike, except in subordinate periods which are in 

 equation (1) in an ascending order of magnitude, and apparently 

 unconnected with each other; and in equation (2) they are in a 

 descending order of magnitude, being connected with a proportion 

 1, i, i J, &c, in order. 



If we suppose that the least common multiple of the subordinate 

 periods K l5 K 2 , K 3 , &c, in equation (1) is the same as K in equation 

 (2), then the two equations are identical except as to the inversion 

 of the order of the terms, and in that some of the terms of (2) 

 whose coefficients are zero will not appear in equation (1). 



6. It might be asked what reasons make us give preference to the 

 form of equation (1) instead of to the form of equation (2) ; the 

 reply is that the form of equation (2) includes all the subordinate 

 periods that are possible whether they may really exist in a particular 

 series of observations or not, and it is therefore confounding and 

 complicated on account of introducing unnecessary terms in the 

 equation, whereas the form of equation (1) contains only all the 

 unknown subordinate periods, that is, those which may really exist in 

 a particular series of observations. To illustrate practically the 

 advantage of the one over the other in point of simplicity, we shall 

 take a case in which a particular set of observations is composed of 

 subordinate periods of 5, 7, and 15 days only. This can be repre- 

 sented by the form of equation (1) by three terms only, while by the 

 form of equation (2) 105 are required, although it will prove, after all 

 the periods are found, that all of them have zero for their value except 

 the terms whose periods are \, y 1 -, of the full period. On the other 

 hand, if a particular set of observations is really composed of subor- 

 dinate periods 105 in number, viz., those whose periods are 105, ^^p-, 



&c, the form of equation (1) will still hold good. It really 

 does not affect the forms but the ideas only. 



Having found the equation for any kind of observations of a periodic 

 nature, our next step is to see what results take place if certain 

 operations be performed on the observed values of equation (1). 



7. Before doing this let us prove some preliminary propositions. 



It has already been established in chapters on the subject of Sum- 



