96 



G. H. KNIBBS. 



10. Illustrations of applications of sine formulae, mar- 

 riage rates.— The application of the formulae for the fluctu- 

 ations may be illustrated by the following examples. The 

 corrected mean of the marriage rates for the Commonwealth 

 during the years 1907-9 is given in the following table : — 



Jan. 



7-52 



Feb. 



8-01 



March 



April 



May 



June 



July 



Aug. 



Sept 



Oct. 



Nov. 



7-65 



10-40 



6-99 



787 



7-23 



7*05 



7-87 



7-30 



736 



Dec. 



813 



If we group these in four-montlily periods, the three 

 averages for these periods are 8*40 : 7*28: 7*67: 

 The frequency is y = a + b sin (#4-/3); and 

 a 



i (8*40 4-7*28 4-7-67) = 7*78 



From equation (L2) we have 



7*78-7-28 



tan P 



8*40 - 7*67 v3 

 Therefore tan P = 1*186 and P = 49° 52'. 



Lastly b = ^-(8'40-7*67) sec p from (13a) 



= '698 x '73 x 1*551 = *79. 

 Therefore the frequency is y = 7*78 4- 0'79 sin (#4- 49° 52') 



A convenient and sufficiently approximate verification 

 may be had by Simpson's rule : for over the first four- 

 monthly period the values of x corresponding to the first, 

 middle, and last ordinates are 0°, 60° and 120°, and the 

 corresponding values of sin (x 4- 49° 52') are 

 sin 49° 52' - '7645 ; sin 109° 52' = '9405; sin 169° 52' = '1759 

 7645 4- 4 X -9405 4- *1759 4*7024 



also 



6 6 



and 7-78 + ('79 x '7837) = 7*78 4- '62 = 8*40. 



7837 



For the case of quarterly results which occur more fre- 

 quently in practice than four-monthly results, the quarterly 

 averages corresponding to the monthly averages given in 

 the last example are 7*73 : 8*42 : 7*38 : 7*60. 



As the sum of the first and third quarters* averages is 

 not equal to the sum of the second and fourth, we must use 

 equations (18) and (19); and we thus get 



