450 
PROFESSOR K. PEARSON ON MATHEMATICAL 
and 1—^1 f^nd + 1 are the roots of 
z- — rs; + e = 0. 
where r and e are to he determined as in that memoir, pp. 368-369, 
We have ; 
gi) (1 + %) 
(xxiii.), 
(xxiv.), 
where 1 — and r are both negative. This gives a.^ Thns 
and from Ecjuation (xxi.) 
■'» “ r (,. - 5 , - ]) r (fj + 1 ) ■ • • 
and a are known, 
(xxv.) 
we find tlie remaining unknown constant for tlie shape of the curve, y^. As before, 
various approximations may be used to the values of the P functions when either 
or or both are large.! 
AVe easily obtain for the distance between mode and mean 
and for the skewness : 
r? = 
»(gl + go) 
f?! - a) (ft - ft 
(ft + ft) s /(gi ~ ft ~ 3 ) 
(ft -ft)\/{(ft - l)(ft + 1)} ■ 
(xxvi.), 
(xxvii.). 
(5.) A special case of some interest arises when the start of the curve is a 'priori 
known. Sujopose its distance from the mean to he c and let (using moments about 
centroid) 
i^:h^ = y- 2 ^ /^ 3 /( 2 ^ 3 c) = Ts.(xxviii.). 
Then we easily find ; 
_ 1 ~ 7i _ _ _ 7] + go _ 
(1 + ft) (— ft + ft + 3) ’ (1 + ft)(!?i — 'p — T) 
* 1 - (^1 Ijeing negative, € is negative, and accordingly by what goes before ko lies between 1 and oo. 
t The value of yo for curves of Type L, if m\ be’small luit m -2 large (‘Phil. Trans.,’ A, vol. 186, p. 369, 
foot-note), is 
Ih 
a. 
h 
{mi 4- ?«2 + 
+ 111-2 
rih 
g 1 2 + ijij 
.1 ) 
■ r(mi + 1) 
and this can be easily modified to suit (xxv.) above. A very convenient and exaet formula for P {n + 1), 
if n be large, is that given by Forsyth (‘B.A. Reimrt,’ 1883, p. 47): 
s/fl- -I- n + Mw+i- 
the error being less than 
1 
240«3 
r(n + 1) = s/ott 
of the whole, 
