STUDIES IN STATISTICAL REPRESENTATION. 



93 



which determine «i and a 5 ; when a x and a 5 can be at once 

 deduced from (39) and (40). Lastly reverting to equations 

 (29) and (29b) we see that 



(43)...(n + r3 + n + ^7 + ^9 + ni) = (r 2 + 1*4 + r 6 + r 8 + r 10 + ri 2 ) = 



a 6 . cos 6 a 6 



12 



7T 



The data are not sufficient to enable us to determine both 

 a 6 and a 6 , but with data thus restricted we may assume a 6 = 

 and we get at once the value of the amplitude a 6 . 



(44)... a 6 = -^ I n + r s +.r B + r 7 + r 9 + r n 



The data at our disposal do not admit of solutions for terms 

 higher than 6x. 



8 Quinquennial periods.— Quinquennial periods with 

 yearly results only are fully represented by two periodic 

 terms. Let T u Y 2 ... be the yearly results and q x , q 2 ... the 

 yearly means, then 



(45)... Y 1 =j \a + b sin (x + /3) + c sin 2 (x +y) 

 and / 



dx 



(46). ..q! = Y 1 /47r = a 



/ 5 



2tt 



b cos 5 (1 - sin 18°) + 



b sin /5 cos 18° + \ c cos 2y (1 -(-■ cos 36°) + \ c sin 2y sin 36° [ 



with four similar expressions indicated in the table here- 

 under. 



If sin 18° be denoted by s x ; cos 18° by c 2 ; sin 36° by s 2 and 

 cos 36° by c 2 ; the coefficients of b cos /?, b sin ft, \ c cos 2y, 

 and J c sin 2y inside the bracket are given by the following 

 scheme : — 



Mean. 



b cos /3 



b sin /3 



J C COS 2y 



J c sin 2y 



9i 



1 - *1 



Ci 



l+c 2 



«2 



ft 



Si +c 2 



s 3 — c x 



— C 2 - Sj 



— S 2 — C]_ 



$3 







-2 s 2 







2 Cl 



$4 



- S l "~ C 2 



s 2 -c L 



C 2 +*i 



— S 2 — C ! 



9s 



-l+s"i 



c i 



- 1 -c 2 



s a 



