378 
Miscellanea 
it occurs in .t- = 3 and y=l, 2 and 3 in Table IV. and consequently in all the first 9 groups 
of Table V. and it is thei'efore correctly treated. 
I have found when dealing with correlation tables that the summation method saves time 
when the table is fairly large. The method might be extended to simplify some of the 
calculations which are required when non-linear regression is investigated (see Professor 
Pearson's Drapers' Eesearch Memoir, Biometric Series, II. On Skew Correlation). 
II. 
The Application of Certain Quadrature Formulae. 
The usual quadrature formulae express an area in terms of the ordinates at the beginning 
and end of the base and a varying number of intermediate ordinates, but in some statistical 
work it is convenient to have values for the areas in terms of ordinates both within and without 
the base on which the area stands. Symbolically we have to express j y dx in terms of 
Vo, yu Vi^ y-'i^ etc. 
Let y.j. = a-\- h.v + cx^ + dx^ + ex^, 
then /'.^•'''^^■*^ = «+]| + ^' 
and ?/o = a, 
?/-i+y + i = 2(a-l-c + e), 
.y-2+3/ + 2 = 2(a-f4c+16e). 
Now assume the required integral can be put in the form 
^yi>-\-k{yi+y -i)+l{y2+y -2), 
substitute the values given just above and equate coefficients of a, c and e respectively to 
1, and and we obtain 
I. J* "^^' = 5^6 {5178^0 + 308 (y, - 17 d/^+y-^)}. 
II. If y = a-\-hx-\-cx'^, 
1 
I _ , 3'<^-* = 2i {3/ - 1 + 22yo +yi} ■ 
III. If yx = a + bx-\-cx^ + dx^ + ex*, 
1 
j . .yxt^'^^ = 5760 {6463yo- 2092^1 + 2298^2- 1132 ;/3 + 2233/4}. 
IV. If yx = a + bx + cx^ + dx^, 
f- 
, yA-^ =-- ^ {27yo + 1 73/1 + 53/2 - 3/3} 
V. If yj: = a + bx + cx\ 
, yxdx = -L {25yo - 23/1 + 3/2}- 
VI. If yx = a + bx + cx\ 
, = ^ {1 3 (3/ _ . +3/p - (y _ I + J/j)}. 
