PROE. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 93 
Thus 
(P '4 "f" (p4p2 5 Pe) (3 p/ 5 PsPg) ~ 
The two roots of this quadratic are clearly and w. 2 , so that the complete solution 
IS 
Cl — a 
P ~ '^1 
ICi — w„ 
CT]^ =: ll ^Wi, 
%0i — IL:, 
Co, = a ---, 
%Vi - W2 
0-3 = h v^'^Co, 
where Wi and w.^ are roots of 
(p4 ^p3^) d~ sPe) ^ (sPr^ sPiPe) — d . . . (36) 
( 12 .) Now we may note several general points about these equations. 
Let Wi be the greater root, then if 
(i.) /to lie between Wi and w^, c^ and C 3 are both positive, or the frequency-curve is 
the sum of two normal curves. 
(ii.) p ,3 > Wi, Cl is positive and C 3 negative, or the greater component group is 
positive, we have then a real difference solution. 
(iii.) /To < 1 ^ 2 , Cl is negative and 03 is positive, or again the greater component group 
is positive, or we have a real difference solution. 
Obviously if = 3 p 3 ^, and coefficients of the quadratic (36) all 
become zero, but these are just the conditions which would be satisfied if the 
frequency-curve were a true normal curve. This gives for all practical purposes a very 
sufficient test of whether a given symmetrical frequency-curve is a true normal curve 
If /Xj. he not equal to and p-g he not equal to then ive have no right to 
assume that a symmetrical frequency-curve refers to homogeneous material. We must 
then investigate whether a better result cannot be obtained by treating it as two 
superposed normal curves having the same axis. 
The quantities 
P 4 — 6p3" 
6po' 
and 
■— 
_Pf, — opjp^ 
(3po 
I propose to call the excess and defect of the frequency-curve. The excess measures 
the excess of one-third of the fourth moment over the square of the second 
moment; the defect measures the defect of the fourth moment from one-fifth the 
ratio of the sixth moment to the second moment.* Here “excess” and “defect” 
are used in the algebraic sense, and may take either sign. They appear to be a good 
* The introduction of the factor into both excess and defect is to preserve a relative as dis¬ 
tinguished from an absolute measure of divergence. 
