PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 79 
H-2 — — f 
H = /s — + 'Iq^ 
+ ^^>2 — 3 ^^ I 
H-5 — ~ + 10^2Vs ~ lO'ZVs + 4^® j 
The centroid having been found, it may be asked : Why we should not calculate 
fio, /xg, {x^, fx~ directly ? The answer lies in the fact that the centroid will not generally 
coincide with a unit division on the deviation axis, and the powers to be calculated, 
instead of being those of two place figures, become in general powers of numbers 
containing three or four figures. Thus the labour of the arithmetic is much increased. 
(&.) Graphically .—If the figure be drawn on a large scale, the moments may be 
found with a fair degree of accuracy by aid of the following process, which has long 
been of use in graphical statics for finding the first, second, and third moments of 
plane areas.* 
It is required to find the moments about O'y' of tbe curve ABC, bounded by the 
straight line O'CA. Take 0"y" parallel to O’y' and at distance li. Take any line 
PP', first to O'y' from AC to ABC; let the perpendicular from P' on 0"y" meet it 
in N', and let O'N' meet PP' in ; let the perpendicular from Q' on 0''y" meet it 
in N", and let O'N" meet PP' in ; let the perpendicular from on 0"y" meet it 
in N"', and let O'N" meet PP’ in Q^. In this manner a series of points Q^, Q^, Q.^, 
Q^, Q^, are detei’inined. Let these points be determined for a series of positions of 
PP' taken at short intervals from C to H, then all the corresponding Q being joined, 
we obtain curves termed respectively the first, second, third, fourth, and fifth moment- 
* The third moment of a plane area is used in determining graphically the moment of inertia of a 
spindle about its axis. The method described is sometimes attributed to Collignon, but seems to have 
been long in use to find “ equivalent figures ” in the case of beam sections. 
