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



Again, from this new series we see that the mean of the two terms 



. nb b 



/ ^\ sm ]T cos 2 



from the first term of the mean series is u sin ( U + — ) and 



\ 2 / . b 

 n sm — 



that the mean of the two terms from the second term of the mean 



nb b 



( n -i-2 \ s * n ~2~ C0S 2 

 TJ + — - — b J , and similarly the mean of the 

 nsin- 



two terms from the third fcerm of the mean series is 



. nb b 

 sm -g- cos 2 



7~b~'- 



n sin — 

 2 



and so on. 



Thus the second mean series will be- 



b b nb 



sin ■ . rcos 7 . n cos ^ 



w sm - n sm - 



2 2 



(. i N sm -77- cos h 

 U+^)_ 2 -_ 2 + ,&c.l . (5). 



n sm 

 2 



Now in these results we see that the following laws hold — 



I. When n is an odd number the first factors of the terms of the 



first mean series are the same terms in order as the ^i-T^th term of 



the proposed series. 



II. When n is an even number the first factors of the terms of the 



(n + 2\ 



second mean series are the same terms in order as the ^— - — Jth term 



of the proposed series. 



III. The second factor in each term of the first or second mean series 

 is a common factor to its own series. 



IV. Each first factor of the terms of the first or second mean series 

 is the middle term of the proposed series from which the correspond- 

 ing mean term is derived. 



Thus the investigation established, that the first mean series is 



