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



Mean series 



= u + K sin (v, + py } { 1 -(^J } 



+ ; % sm(Y 2 +P& 2 )}{l-^-y}+ 5 &c. . . (7). 

 Remaining series 



= { % sin (Y 1 +P& l )}^-Iliy+ {^ 2 sin (V 2 -hP5 2 )}^^J +j &c . (8) . 



Throughout these three expressions a particular value of P will 

 correspond to a particular vertical column, the next greater value of 

 P to the next vertical column, and so on. 



19. We have up to this point derived these last two results as general 



with respect to the factors of the type — which depend on the value of 



q 



m, and of this we have only said that it is an odd whole number. 



Oiir next step is to see what conclusions we shall arrive at when we 



apply some particular value to m. 



(1.) Suppose that mc is less than K 1? therefore it is also less than each 



of K 2 , K 3 , &c, and m\, mb 2 , &c, are each less than 27r ; thus the factors 



• mb, . mbc, 



sin — i sm — * 



2 2 11 

 — , — , &c, or their equivalents — , — , &c, are positive. 



msin-i msin-? % l ^ 2 



2 2 



We have shown before that each of the above factors is a proper 

 fraction, and consequently each of their equivalents is a proper frac- 

 tion ; thus each of gr x , q 2 , &c, is greater than one, and therefore the 



value of each of the factors ( ilZli^ , ( , &c, when r is suffi- 



V 2i / V It J 



ciently great reduces to zero. 



(2.) Suppose that mc is equal to K 1? therefore it is less than each 



of K 2 , K 3 , &c, then, as before, we can show that — is zero, and each 

 of q 2 , q s , &c, is positive, and the value of ( ^3 — or ( 1— - \ is one, 



\ 2i J \ aJ 



and the value of each of the factors — -Y, — -\ , &c, when r is 



\ % / \ 23 / 



sufficiently great, reduces to zero. Thus, when mc is equal to the 

 period K l5 and the number of operations of taking successive means 

 and remainders are sufficiently performed, the remaining series will 

 be reduced to u Y sin (Vj + PZ^), or a subordinate series in the observa- 

 tion whose period is K x . 



(3.) Suppose that mc is greater than K x and less than 2K l5 and 



