Periodicities ivhen the Periods are unknown. 373 

 = Ul sin (Y x ) ( J±^P? +u 2 sin (Y 2 ) + , &c. 



2l 22 



1st mean from the rth set of means 



=i h sin (Vj) (fr- 1 )*" 1 ^ sin (Y 2 ) f , Ac. 



1st remainder from rth set of remainder 



=Ui sin (Y x ) ^l^- r +^ 2 sin (Y 2 ) fc±> r , 

 2i r 22 r 



and similarly the corresponding numbers from the second vertical 

 column will be represented by the following expressions — 



2nd observation from the observation series 



sin (V 1 + & x ) -f- u 2 sin ( Y 2 -f Zj 2 ) + , &c. 



2nd mean from the first set of means 



= u-\-u 1 sin (Vi + b{) — + n 2 sin (Y + 5 ) — -f , &c 



2i *■ 22 



2nd mean from the second set of means 



=i h sin (Y 1 + il^i + u 2 sin (Y 2 + &,) +, &Q. 



2i 22 2 

 2nd mean from the third set of means 



= Ul sin ( Vj + b{) (gl ~ 1)2 + «, sin (Y 2 + 6 2 ) ^~ 1)3 + , &c. 



2i 22 3 



2nd mean from the rth set of means 



= Ul sin (Y 1 + & x ) — + w 2 sin (Y 2 + 6.) fe" 1 ^— + , &c. 



2/ " 22 r 



2nd remainder from the rth set of remainders 



= i h sin (Y x + ^~^* + u, sin (Y 2 + b 2 ) (lnR r + , & c . 



2i 22 r 



And the corresponding numbers from the (jp + l)th vertical column 

 will be represented by the following expressions — 



(p + l)th observation from the observation series 



=u+u 1 sin (Vi+pbi) ^i^sm (V 2 +pb. 2 ) +, &c. 



(j? + l)th mean from the first set of means 



=w + **i sin (Yi+pbi) — + %sin (Vo+jpb 2 ) — + , &c. 



2i 22 



