78 PROF. K. PEAESOX ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Hi = Hi + T j 
H3 = ^.(2)- 
Hi — Hi + 
H5 = -"^'iH i + 10(//x'3 + ^ 
(iii.) The distance of the centroid of ABC from y'y' is the ratio of its first moment 
y.\ha to its area a, and = y-' 
(iv.) To find the successive moments of a given curve about a given line. 
For the purposes of the present problem we require only the first five moments of 
a curve like ABC about a line yy passing through its centroid. The solution may be 
obtained either analytically or grajDhically according to the accuracy or rapidity with 
which we wish to work. 
(n.) Analytically .—Suppose the frequency-curve to be obtained by plotting up the 
results of 1000 measurements, each unit of length along AC corresponding to an 
equal change in the deviation. Starting from the point C, beyond which no 
individual occurs, we may have in practice, perhaps, 20 to 30 equal ranges of 
deviations before we reach the point A, which terminates the deviations on the left. 
The equal range being taken as the unit of length, let the numbers in the groups at 
1 , 2, 3, 4, 5 . . . units of distance from C be y^, y^, y^, y^, 2/5 . 
Then the moment clearly equals very appi'oximately 
1" X 2/1 + 2 " X 2/3 + 3'' X 2/3 + 4" X 2/r + . . 
or since a — 1000 , and h may be conveniently taken = 100 , 
, _ 1“ X 2/1 -I- 2“ X 2/0 + o'* X 2/3 + 4“ X 2/4 + • • • 
^ ~ 100“ X 1000 
Sufficiently accurate values can then be found for ph, p'g, p'^, ph, provided we 
know the 2nd, 3rd, 4th, and 5th powers of the natural numbers up to about 20 to 30. 
The values of these powers up to 30 are given later in this paper. 
Knowing the first five moments about the vertical through C, we can find the 
centroid by aid of (iii.) above, and then the moments about the vertical through the 
centroid by aid of equations ( 1 ). 
Since pj = 0 for the centroid p\ = y, and therefore we have the folloving to 
determine the other moments :— 
