86 G. H. KNIBBS. 



Since the form in which the data are usually furnished is 

 that of average values for a period, e.g., monthly, quarterly 

 etc., the data in question are not given in the form of 

 equation (11) but in the form 



Y/fdx = y; i.e. ¥/&) or F/(i*r) 

 for the cases of monthly or quarterly statistics as the case 

 may be. 



Going back to equation (10a), it is seen that, if the data 

 furnish values for half years only, then y = Y/ir, and a is 

 the mean between the two^values of y': also either b or P 

 may be arbitrarily assumed and the other then deduced. 

 Ordinarily it will suffice to put /? = 0. 



If, however, results are given for thirds of years, a unique 

 solution for a, b and P, mayjbe obtained. Integrating (10a) 



between the limits to ~, ~ to ~, and — to 2?r, the 



group results are : — 



|Yi = f 7r a + J b (3 cos P + v3 sin P) 

 Y 2 = f ira + \ b ( - 2v/3 sin;)3) 

 Y 3 = f Tra -f jf b ( - 3 cos /? + v3 sin /3) 



Whence dividing by J rfa? or| -^ we get, in each of these 



'o <, * 



expressions t/' k instead of Y k , a instead of 2^/3 and 3b/4?r 

 instead of \ b.\ 



Hence primarily, by addition \ 



(11) 3a = y'i + y' 2 + y' 3 ; thus a = ^(y\ + y' 2 + y\) and 



3v3 

 ri9\ l/ r i-2 ^ 2 + y'z __ a-y'* _ 2tt bsinft __ 1 r q 



K } z(y'i-y's) y'l-y's JL 'bcosP ^3 



2^r 

 and, P being found, we have 



(13) b = — (a - t/ 2 ) cosec P ; or 



(13a) b = |pfoW,)sec£. 



