370 Mr. V. N. Nene. On a Method ot Tracing 



+ % sin (JJ 2 +^P b>) sin (U 3 + ^ & s >— + , &c. 



{ n - ( m _i) sin | Ui+ &i j qvzl 



+ , &c. 



Let these remainders be entered in the third line of our table. 



Again, if successive means be taken of m of these remainders 

 precisely in the same manner as the means of the observations which 

 were taken before, we shall have a set of second means, and similarly 

 if these means be subtracted from the first set of remainders we shall 

 have a second set of remainders ; and, further, if these operations be 

 continued r times we shall have an rth set of means and an rth set of 

 remainders, and the rth set of means taken in order will be equal to 

 the sum of the terms of the following expressions* — 



1st mean =7^ sin (jj-, + — — - rb->\ — ^ 



+ He m(v t +^rb t y**- 1 }-+, &c. 

 2nd mean = 7^ sin (jj-, -f — — - rfr-, + b^\ — ?^ — 



+ u 2 sin (u 2 r& 2 + & 2 ) ( -*^-+, 



3rd mean=w 1 sin ( Uj + -— r\ + 2\ \ 1 



V 2 / 2i r 



+ % sm/U 2 +^r& 2 + 25 2 )fe- 1 ) r - 1 + , 

 V 2 / a/ 



&c. 



{._^-l) r j th = , lS in l^ + ^-^r-l^^j^r 1 



+ w 2 sin | U 2 + (™- m ~ l & 2 J ^— ^ + , &c 



And similarly each of the rth remainders will be represented to be 

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



* See two next paragraphs. 



