88 G. H. KNIBBS. 



and since 



(17)... /{a + b sin (x + /3)j dx = ax - b cos (x + /?) 



the following are the values for the integrals between the 



limits indicated, viz., for the case where sum of first and 



third quarters equal sum of second and fourth. 



Yj = \ira + b (cos P + sin P); Y 2 = -| ira + b (cos/? - sin jG) 

 Y 3 = | Tra - b (cos /? + sin /?); Y 4 = J ^a - b (cos /? - sin P) 



Let |/' k = Y'k/iK in which fc = 1, 2, 3, and 4, and g\, t/' 2 , 

 t/' 3 , and t/' 4 are the corrected quarterly means. Hence in 

 these expressions we may write t/' k instead of T k ; a instead 

 of \-*a ; and 2 6/tt instead of b and we thus obtain 

 (18)... a = i (y\ + t/' 2 + i/'g + f/'O; 

 and also 



8b • R 



, , , — sin p 



(19)... *r»rV +v , =i = tan p 



IT 



and p being thus found, we have 

 (20)... b =~(y'i-y'2-y' 3 +y' i ) cosec P 



o 



= -g (l/'i + lA ~ S/'s " y'*) sec /? 



If, as is very commonly the case, the sum of the first and 



third quarters is not equal to the sum of the second and 



fourth, we must introduce another term into the frequency, 



say c sin 2 (x + y); then making c = b in accordance with 



the arbitrary assumption already referred to, the frequency 



becomes 



y = a + b sin (x + P) + b sin 2 (a? + y) 



Proceeding on the same lines as before we have 



2 2 



y\ = a + — b (cos /3 + sin 3) + — b cos 2y 



7T 77 



2 2 



y' = a + — b (cos /3 - sin 3) - — b cos 2y 



2 2 



y' 3 = a - —b (cos 4 sin 3 )+ — b cos 2y 



TV TT 



2 2 



^/' 4 = a — — b (cos /3 — sin /3) — — b cos 2y 



7T ~ 



