Periodicities when the Periods are unknown. 375 



each case gives the value of the first, second, third, &c, numbers 

 respectively to the right of the first strong vertical line. 



17. We shall now take the sum of all the mean series and put the 

 result by one series only, thus the sum of all the mean series is equal 

 to— 



sin (Y/+PM} I ~ +^ + ( -^P- .... iiiriT! ) 



I 2i 2i 2i 9.1 J 



+ K sin (V 8 + P5 2 ) } { 1 + i^- 1 + toz^2! .... ( ft- 1 )"" 1 j 

 L 22 22" 22 2/ J 



+ K S in ( V 8 + P6 3 ) } { 1 + 23=_1 + (jr=pi .... (Ss-lt. 1 ) + , 4c. 



I- 23 23 23 23 J 



Suppose S=l + £^+(2^(2zdi\...(^ir; 



2 2 3 r 2 4 2 r 



therefore 



Hence by subtraction si=i ( 1 — - — X , 



by division S = l 



=l_/2-l 



Cr)' 



18. From this result we can write the sum of all the mean series — 



u + { , h sin ( V, + VbJ } { 1 - (&=± J } 



+ {« 2 sin (V 2 + PJ 2 )}{ 



Thus from one series of observations we get two series, one is the 

 sum of several mean series, and the other the series of last remainders, 

 and of course the sum of these series is equal to the observation 

 series ; thus whenever we want the sum of all the mean series we 

 shall get it by subtracting the last remaining series from the corre- 

 sponding observation series. We shall rewrite these series below and 

 call them observation series, mean series, and remaining series respec- 

 tively. 



Observation series 



==% + sin (V x + Vbj) } + {w 2 sin (Y 2 + P& 3 ) } + , &c (6). 



