382 



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



slightly different from the preceding ones : we shall, therefore, first 

 of all apply them to the general form and then to particular examples. 



31. ' Suppose thatmc=K w where K w is a known period and mis some 

 integral number. Then using m as working value let us find as 

 before the first mean series from the observation series, the first series 

 of remainders, and the second mean series ; but instead of subtracting 

 the terms of the second mean series from the corresponding terms 

 of the first series of remainders to get the second series of remainders, 

 let us add them in the same order and call this series the first 

 difference and sum series. Again from this series let us find, in the 

 same manner as stated above, the second, the third, &c, and the 

 rth difference and sum series. The rth difference and sum series will, 

 as will presently appear, be of the following form — 



^sin (Vi + P&x) (l-^j + uz sin (V 2 +P& 2 )^1— ij+ *. . . . 



u n shi (V n +Vbn)( 1— lY+w„ +1 sin (V +p^ +1 ) ( 1__LY+, & c . 

 V 2«/ V 2»+i/ 



.... (9). 



32. The following calculations are made for ready reference and to 

 show how the factors of the form (l — — ^ , which occurred in our rfch 



difference and sum series, are obtained. 



The factor for 1st means =-. 



2 



The factor for 1st remainders =^1— i^. 



The factor for 2nd means =/l — ^\ 



V 2/2 



The factor for 1st difference and mean series 



-('-?)? 



The factor for 3rd means = 

 The factor for 3rd remainders 



K'-?)-H)rH)H> 



The factor for 4th means = ^1 — -L^ 



1\1 

 2' 



