84 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Write 
^4 = — 3/x^, X 5 = SOP'iPs — 3/^5.( 22 ), 
and put 
lh=PiP-2 .(23), 
then, iimltipljing up, the above equations become 
‘^PP ~~ H~ ^Pi" — b.(24), 
5/^3T3 - 2^3^ + ^IhVi - 20p3p2® - hpi = b.(25). 
From these equations let us first find^^g in terms of^L. Multiply the first by ^3 and 
subtract from the second 
^PiPi + P3 ( 4 /^ 3 " + \lh - 2p/) - 20 ^ 3 ^ 3 / _ = 0 . . . (26). 
Multiply (24) by 2^3 and add to (26) we find 
2P3^ + P3 (- 4 /X 33 + \^2h - 2^2®) - ‘^p-iKpi - KP-^ - = b, 
or 
„ _ / 97 ^ 
W - + ipi’ . ^ 
Hence, so soon as 'Pz known, = P-d/'Pz can be found, and then and yo will be 
the two roots of the quadratic 
y^~Piy + P2=^ .. • (28). 
Returning to (27), substitute this value of yjg in (24), and we have an equation 
containing ^92 only, on which the whole solution of the problem now turns. 
This equation is the following one ;— 
24^3.'^ - 28\,P27 _p 36/^3^^° - (24ia3X3 - IbX/)^^^® “ + 2 X 5 ") pP 
+ (288^13*- 1 2X3X3^3 - x/)ft= + (24/,33X3 - 7iJ.,-K-)P-i + 3 inXP-, - 24^' = 0.(29). 
( 8 .) Some remarks may be made on this equation. Since this equation is of an odd 
order, one real root may always be found. Further, remembering that X^, = — 3jaj, 
and Xj = 30p.^,p3 — 3 / 4 . 5 , we see that in the case of a normal curve, for which 
Pi — 8 /r/, while /Tg and p,^ = 0 , all the coefficients of the above equation of the 
ninth order vanish except the first. 
Thus as we should naturally expect, will be zero. Accordingly, since, with 
increasing symmetiy, the coefficients become small, it will be needful to work their 
values out to a greater degree of exactness the slighter the degree of asymmetry. 
