Miscellanea 
379 
VII. If y^--=a + h.K + cx^ + d.i^->rex\ 
1 
j - . ^^'^•^ = 1440 f®^^ - i +^+ i) - - i + ^i) + 1 1 (y _ 5 +y j )). 
VIII. If y:, = a + h.r + cx'''-»rdx^ + ex^, 
IX. If = a + 6.^(7 + c.»^, 
The most obvious puriiose for which these formulae can be used in biometric work is 
for calculating the areas from the values of ordinatcs when the equation to the curve has been 
determined. It is clearly convenient to calculate as few ordinates as possible and I have found 
Formulae I. and II. especially convenient when expressed in the form of difi'erences ; thus 
II. can be written 
j _^ydx=y,, + ~{Ay„-Ay_-i}, 
and the adjustment from the mid-ordinate is easily calculated. 
Formula I. can be written 
ri 291 17 
and the fractions are -0505 and -00295. 
Formulae VI. and VII. are useful when the ordinates at the beginning and end of each 
group have been calculated. 
A numerical example of the use of these formulae in a case where their accuracy can 
be tested will be of interest. We will assume that the ordinates of the normal curve have been 
calculated for the equidistant values of x and the areas corresponding to these as mid-ordinates 
are required. 
TABLE VI. 
X 
Value of 
ordinate 
A 
Area by 
Formula I. 
Area by 
Formula II. 
True value 
•80 
•2896916 
•0785785 
•40 
•3682701 
•0306722 
•366223 
■366274 
•366218 
•00 
•3989423 
- ^0306722 
•396307 
•396386 
•396298 
•40 
•.3682701 
- ^0785785 
•366223 
•366274 
•366218 
•80 
•2896916 
- ^0955055 
•288992 
■288986 
•288994 
1-20 
•1941861 
- •0832653 
•194740 
•194696 
•194746 
1-60 
•1109208 
- ^0569298 
2-00 
•0539910 
The second and the last columns of this table were found from W. F. Sheppard's Tables, 
Biometrika, Vol. i. pp. 182 et seq. For frequency curve work we do not generally require a 
large number of figures, and as the total frequency does not often exceed 5000, Formula II. would 
be sufficiently accurate while Formula I. would give a satisfactory answer in almost all possible 
cases. 
48—2 
