xvi Tables for Statisticians and Biometricians 



or, restoring h, 



h 



5760 



308 



_i) - 17 (y s+2 



(4), 



and similarly by integrating from ^ to f and from | to f , we find : 



f 1387/ 8 4- 68y s+ i 17y s+2 } 



138y g + 5348y m + 223y a+a } 

 ! + 2298y s - 1132t/ 8+ i + 223y g+2 

 |223^ a - 1132^ + 2298y g - 2092y s+1 + 6463y 8+2 ] 



5760 1 

 h 



...(5). 



It is often better, especially when the curve has a finite terminal ordinate, to use 

 these values rather than to put extreme y's zero in (4), when we are dealing with 

 terminal frequencies*. 



Another origin, i.e. at the start of the five frequencies, is often convenient, for 

 example in finding, not the mid-ordinates, but the bounding ordinates. If we take 

 as our biquadratic y = a + 2bx + 3ca^ + 4cfe 3 + o&e 4 as before, then by integrating 

 from to 1, 1 to 2, etc. we find 



! + 274n, - 126w s+1 + 



(6). 



lifts-! + 15( 9w s +j + ! 



4 ., , - . 4w g _! + 6?i s - 4rc g+1 + n s+2 ) 



From this quartic equation, by putting # = ^, f, etc., or from the previous 

 equation, by putting #=0, 1, 2, etc., we obtain, on the other hand, the mid-ordinates 

 in terms of areas, or frequencies, as follows: 



v -116(7i 4 ._ 1 + rc s+1 ) 



1920A 



{9 (n s _ 2 



s - 2= 1920/^ 1689Ws -- 2 

 1 



1920/i 



1 

 : 1920/t 



:_ 1 -746/7 s + 364n s+ ,- 

 2044w s _! 26w s 30w s +i + 9n, 

 - ZU'14'W s _|_i 7 l?v s -f-2 

 - 746n s + 684?7 8+] + 1689w 



,+ a 



(7), 



...(8). 



The results in (8) are convenient at the tails, where it is not always desirable to fit 

 the biquadratic to one or more zero frequencies beyond the actual range of frequency. 

 * See, however, a paper by E. S. Martin in Biometrika, Vol. xxin. 



