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



3rd means = — — 



3rd remainders^ 



q z q 6 q 6 



_ (g-D»( g -i) _( g -i)» 



q s q* 



rth means — — 



rth remainders 



_ ( q -iy-\ ( q -iy-i _ ( q -iy-i q -( q -iy-i 



q<~ 1 q' q' 



= (f-l)'-'(j-l) _ (8-l)' 



q' f 



15. We mast notice that at each operation of taking means we. 



lose — — - values at each end of the series, thus the rth mean series 

 2 



commences in our form after — — \r observations. For this reason 



2 



suppose strong vertical lines are drawn in our form at the ^ — 32r th 



2 



observation from each end, and let us confine our attention hereafter 

 to the series embraced between the two strong vertical lines. 



16. Then, of the numbers occupying the first vertical column from 

 the first of the strong vertical lines mentioned above, the following, 

 viz., an observation, an individual of the first set of means, an indi- 

 vidual of the second set of means, an individual of the third set of 

 means, &c, an individual of the rth set of means, and an individual 

 of the rth set of remainders, will have for their mathematical equiva- 

 lents the following expressions, taken in order, viz., 



1st observation from the observation series 



=u +«i sin (Vj) + u 2 sin (V 2 ) -f , &c. 



1st mean from the first set of means 



=u + u 1 sin (Vj) — + u 2 sin (V 2 ) — + , &c. 



1st mean from the second set of means 



= uy sin (Vj) +u 2 sin (V 2 ) + , &c. 



1st mean from the third set of means 



