Karl Pearson 
7 
Expanding and collecting terms in powers of //, we have 
log, U ={{p + q){l+ p) + l] log, a + p)- {(p + 7) P + 1 } P - p( l+p) "^.prf 
1 P + q 1 + 2p p'^ + q-^ 
p{l+p) pq 2p'{l+pf fq" 
+ ^p^l + pry .f) p^i + pT Ir <f'^ 
1 - -P 
= {« (1 + p) + 1 ! i..gv (\ + p)-[,>p + i] log, p - yi ^1 ^ - 
+ etc. 
1 + 2p Q -J _ I4^3p + 3£'^ - P ^ 
^ii?{\+pf 'P'Q; m'{l + pf P'Q' ) 
Hence, putting a- = m (1 + PQ, 
log. « = {n (1 + P) + 1 1 log,. (1 + p) - ( + 1 ) log,, p - ^ ^ V^TVirfv^^ 
1 //^ ^ _ 1 + 2p i^PQ\ 
Similarly 
1 / l + 2p Q^-P _ 1 + 3p + 3p-^ ((^ -P)(l -QP) \ ^ ^^^^ 
6 Wm Vr+p VPQ (1 + pf (PQ)* / 
1 1 \ h l + 2p P-Q 
12Py m(l + p) I2a ,j^i + pf (PQ)?} 
/r l + 3p + 3p' U-3PQ 
12<r'^ mHl + r)" PQ • ■ 
If we stop at terms in — we see that log, y contributes only to the constant 
m 
term and this may be included in G. Into the same foctor C may be put the 
logarithmic terms in log,,?/. Thus finally we have* : 
Ih Q-P^ 1 Ir l + 2i, Q-P 
We note the following facts with regard to this result : 
(i) The series converges with a factor Ij^/m. Unless this be small we cannot 
neglect the terms in h and h\ In other words deviations in excess and defect of 
mP are not equally probable. Thus skew frequency rather than the Gaussian 
hypothesis is indicated from the very start of the investigation if our second sample 
be not considerable. 
* The existence of the term in (/(/cr)-' of the same order as that in /i/cr prevents us treating the result 
as a normal curve witli sliifted centre. 
