XV 



(52/0,0 + . p i98/i. - 26/_ M - 2 

 {'>2/o,o + 598/0, _! - 26/ M - 2 

 + 598/_ M - 26/ 1>0 - 2 



._, - 23 (/_,,_, 



- /, , 



If the base unit of area be h x k, then the right-hand side of these equations 

 must be divided by hk. 



The above are useful, but of course only approximative methods of passing from 

 ordinates to volumes or from volumes to ordinates in the case of frequency surfaces. 



(ii) Univariate Frequency. 



Here we use a biquadratic for five consecutive points, and find the constants 

 by equating the areas on five equal subranges h. Let the areas or frequencies be 

 H-2> n^, n 8 , n g+l , n s+2 and the mid-ordinates be y g - 2 , y-i, y, y+i, y,+i- They are 

 represented in the accompanying diagram. 



ORDINATES AND AREAS. 



If we take the biquadratic to have its origin at the midpoint of n t , i.e. at the 

 foot of y s , then if its equation be : 



y = a + 2bx + Sen 2 + 4-da? + oeu; 4 (2), 



we have 



c = 7V I 16 (y+i + 2/-i) - (l/x+z + y-2> - 30} y, 



Then by integrating we have, treating h as unity, 



+l 



r+l 

 =| 



