80 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
curves. Let the areas AQ^L^C, AQAL^C, &c., he read off with a planimeter, and be 
ttj, a 3 . . . Then 
11 
1 
I 
yd 
= ag/a 
y-s 
11 
I- 
yi 
« 
II 
y-o 
= «5/« 
-- 
A good draughtsman will construct these curves with great readiness, and if on a 
sufficiently large scale, the results may be read to within the one per cent, error."' 
Equations (4) then enable us to complete the problem of finding the moments 
about a line through the centroid. Or, the first moment being found about 0'y\ and 
so the centroid determined ; we may shift O'y till it passes through the centroid, 
and then proceed to find . . . y- directly in the above manner. In this case care 
will have to be taken in reading the areas of the moment-curves, which have now 
pieces of their areas negative, to carry the planimeter point, in the proper sense, round 
their contours. 
(5.) Properties of the prohahility-curve. 
Let the equation to the probability-curve be—■ 
( 6 ). 
Then o- will be termed its standard-deviation (error of mean square). c is the 
total number of units measured, or the area of the probability curve. 
(i.) To find the second and fourth moments of the probability-curve about the axis 
of 
Let them be Mg' and M^'. 
Then 
Mg' = 2 [ y:P dx = c X cr. 
Jo 
M^' = 2 [ y'x^ dx = c X Serb 
y = 
CTv/ (27r) 
-:c2.(2o'2) 
Clearly M 3 ' and M 5 ' are zero. 
[* My demonstrator, Mr. G. U. Yule, has graphically calculated the first four moments of a number 
of statistical frequency-curves, with the object of fitting them to the generalized probability-curve (see 
footnote, p. 74). The method is sufficiently accurate in practice, and I hope soon to have an instrument 
to construct these curves mechanically, designed by him.—February 9, 1894.] 
