K. Pearson 
269 
d fx 
+ etc (iii), 
a'" 
where 6 lies between 0 and 1, and j-^ (6x) = R, say, represents the remainder 
after n terms of Maclaurin's expansion. 
Substituting in (ii) and rearranging, it becomes 
+ 
dR\ , ] 
da. 
dx> Ba., 
+ {/<2'-y'>(r70+5f)H^''' 
+ 
0. 
But the quantities a„, a,, a.^, ... a„_i are at our choice, and therefore to satisfy this 
equation, their variations must be independently zero. Thus we have the following 
equations to ^nd a^, a^, ... a„_i : 
.(iv). 
(y-y') 
a? 
1.2.3 doij 
+ ^^]dx = 0 
Now let A be the area, A/ii, Af^s, Afi^, ... etc. be the first, second, 
third, fourth, etc. moments of the theoretical curve, and A' be the area, A'jx-^', 
A'fi^, A'li-i, A'/jL.^', ... the like moments for the observation curve, moments 
being taken roimd the axis of y (which is of course any axis). Then the above 
equations may be written 
