94 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
measure for practical purposes of the divergence of a given symmetrical frequency- 
curve from the normal type. 
We may now express the quadratic (36) in terms of e, and e^, and analyze the 
results according to the character of the excess and defect. 
The quadratic becomes 
8ei ^ — €3 — + £3 — 3 e^ (1 -f e^) — 0. 
This gives 
W _ £3 ± \/{(eo — 6 ed" + 366^} 
6 ei 
(37). 
We have the following cases : 
(i.) and £3 both positive. Then the values of w are both real, but they must 
also be both positive, otherwise cr^ and 0-3 would not be real. It is necessary, there¬ 
fore, that 
+ 36£i®}, 
or 
£3 < 3 £i (1 -f £1). 
(ii.) and £3 both negative. Then iv will be real if, when 
(— £ 3 ) does not lie between 
and 
If 
then we must have 
V {— ^ 1 ) < 1 . 
6 (~ ^i) (1 + \/(~ ^ 1 )} 
6(-^i) {1 - v/(- ^i)]- 
( — eo) > 6 (— £i) {1 + y/(— £i)]. 
Further, in order that w may have both values positive, we must have 
(-£ 3 )> (-e3-6(-£,)}^-36(-£,)^ 
or 
(— £ 3 ) > 3 (- £i) {] — (— £ 1 )]. 
This latter condition is clearly satisfied if 
(— £]) > 1 . 
On the other hand, if 
\/ (— Cl) < 1 , 
