Periodicities when the Periods are unknown. 869 



observations in order. Now let successive means be taken of m obser- 

 vations (m being an odd integer) advancing to the right by one step 

 at each operation, and entering the respective means under the 



, , &c, terms of the observations, thus forming a second 



2 2 



line of our table ; the means in our second line taken in order will 

 be equal to the sum of the terms of the following expressions — 



1st mean=w + « 1 sin \JJ 1 -f-— — ^b-, \ — + u 2 sin(^U + ^ — - & 2 ^ — 

 V 2 / q 1 \ 2 / q 2 



+ % 3 sin(U 3 + — — -b s ) — + , &c. 

 \ 2/^3 



2nd mean^w + 7^ sin(u i + m ~^~^ 'b 1 \ — + w 2 sin U 2 + ^-t^6 2 ^ — -* 

 \ 2 / 2i " \ 2 J q 2 



+u s sin ^U 3 +^|-^ 3 ) Y + > &c - 



3rd mean = w + sin ( XL + 6 n ^ — + sin flL + m ^~% 5 \ 2L 

 \ 2 / & \ 2 7 2 2 



4 % sin fu 8 + ffl ~^ 63") — + , &c. 

 \ 2 / <7o 



71 — (m,— l)} th =7t + 7i 1 sin 



l~TT O- L m + 1 Ul 1 



L 1Il+ r"-s--pJft 



+ 7/0 sin \Jc+-n— >b 2 — -f%sm \J 3 + <n — — — >b s — + , &c. 



L * 2 J J q 2 L l 2 J J 23 



If these means which are entered be subtracted from the observa- 

 tions column by column, then the remainders will severally be equal 

 to the sum of the terms of the following expressions — 



1st remainder =7^ sin /lT l + 1 ^ — - & — ^ + u sin ( U 2 + — — - ^ — — ~ 

 V 2 / q 1 \ 2 J q 2 



j-u s sin^U 3 + !^^63^ ^zJ"-f , &c. 

 2nd remainder =7^ sin ^TJ 1 + ^li-^6 1 ^ ^ — ?" + u 2 sin ^U 2 + ^ 2^—- 



+ 7*3 sin ^U 3 + 7 -^i-6 3 ^ 



3rd remainder =u x sin /'u i + ---i-5 0^ £l — 

 V 2 / 21 



23 



23 



2 c 



