366 



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



The second factor is positive when nb lies between 2^(21$) and 

 2tt(2N + 1), and b lies between tt(2W) and tt(2N' + 1), or when nb 

 lies between 2tt(2N + 1) and 2tt(2N + 2), and & lies between tt(2E"' + 1) 

 and7rf2N' + 2). 



The first factor is negative when nb lies between 2tt(2N + 1) and 

 2tt(2N + 2), and b lies between 2ir(2W) and 2^(2^ + 1), or when nb 

 lies between 2tt(2N) and 27r(2JSr + l), and b lies between 27r(2N' + l) 

 and 2tt(2N' + 2), 



The second factor is negative when nb lies between 2^(21$ + 1) and 

 2tt(2^ + 2), and b lies between tt(2W) and 7r(2N"' + l), or when nb lies 

 between 2tt(2N) and 2tt(2N + 1), and & lies between ?r(2N' + l) and 

 7r(2N' + 2), where N" and W stand for zero or any integral number. 



nb nb b 



sin —y sin -g- cos g 



11. To trace the changes in the factor — or as the 



. b . b 



n sin - sm - 



2 2 



angle ?i& varies between and 2tr, between 2tt and 4*tt, between 47r 

 and 67r, &c. We shall discuss this subject in a particular case only, 

 viz., when b<2w, as there is no necessity of discussing all the cases 

 for our future purpose. 



We have already shown that when &==— , or when nb — 2ir the 



n 



factor has the value zero, and it is easy to see that when nb=4w or 

 = Qtt or = 2N"7r, where N" is some whole number, the factor has also 

 the value zero. 



We have now to find what value of nb makes the factor a maximum 

 when nb<27r. Put —=x and therefore n=— , thus the first factor 



2 



becomes- 



sin x b sin x 



2x~7~b~n . b ' 

 — sm - 2x sm - 

 b 2 2 



let u' = J^i n j^_ > remove constant factors, 



2x sin - 

 2 



thus 

 then 



sm x 



x cos as — sin x 



dhi sin x 2 cos x _j_ 2 sin x 



dx* x a? 2 x 6 



If we put — = we obtain tan a; = a;, therefore x=0, and when £=0 



