STUDIES IN STATISTICAL REPRESENTATION. 87 



In practice, however, results are only rarely given in 

 thirds of years. Usually they are quarterly or monthly. 

 In the former case, if the sum of the first and third quanti- 

 ties equal the sum of the second and fourth, we may suppose 

 the frequency to be of the form 

 (14) y = a + b sin (x + /?) 



The data do not allow of the inclusion of more terms 

 except by making arbitrary assumptions which can in 

 general have no validity. [If for example in equation (14) 

 we add one term so that it becomes 



(14a) y = a + b sin (x + f3) -J- c sin 2 (x + y) 



we should have only four equations to determine the five 

 unknowns a, b, /?, c, y, and we could obtain a complete 

 solution only by making some assumption such for example 

 as b = c or P = y or that y = 0.] In the case of monthly 

 statistics the frequency may be supposed to be of the form 

 (15).. .y = &q + «i sin (x + a x ) + a 2 sin 2 (x + a. 2 ) + 



a-3 sin 3 (x -f a 3 ) + the final term being a 6 . 



It is first of all necessary that x should either be expressd 

 in rates (calculated correctly for equal periods), so that the 

 numbers may be independent of a secular change taking 

 place during the period under review; or that the absolute 

 quantities should be reduced to a common datum. For 

 example, if the absolute number of births during a period 

 is given, it must be reduced to birth rates or corrected so 

 as to express what it would have been had population 

 remained constant during the time under consideration: 

 so that for example the final formula for absolute num- 

 bers will be of the form 



(16) Y=Pe ( f >it) {a +a 1 sin (x+a 1 )-\-a 2 sin 2 (x + a 2 )+ etc.} 



Y being the absolute number of persons, and P being the 

 absolute number for x = 0. 



From (14) above we have for quarterly fluctuations 

 1 3 



