A. Ritchie-Scott 
111 
= ^ {Adm^. + Bdiii.-^ - dm^^iY 
pa 
= ^2 (« + c) + B^ (a + 6) + « + (2^5 - 2^ - 2B) a - 
= {A + B-\f a + A\ + - (84). 
But 
^P=A{a + c) + B{a + 'b)-a={A + B~\)a + Ac + Bh + (i. d (85) 
and a + h + c + d = N, 
Further (- ,P) a + (,P) 6 + (,P) c + (- „P) 
= a {iV (^ + P - 1) - „P} + 6 (PiV - „P) + c {AN - „P) + rZ (- „P) 
= iV {(.4 + P - 1) a + P6 + Ac) - N^P 
= N{A{a + c) + Bia + b) - a] - N^P 
= N,P-N,P=0 (87). 
The above form (86) of the square of the standard deviation (omitting factor) 
is interesting as involving only the squares of the P's. Since the P's are connected 
by the relation (82) bis and (87) their values may be determined from any two of 
them. 
The P's. 
Since - Xst Sr,« = hP,t 
and - XsrS»"5'i' = SP^r, 
XstXs't'^^'st^r.t- = 8P„8P,'t' 
and ' XstXs't'<yst<ysrRst.,'t' = db i^Pst^Psr) = Sst.s't' 
In conformity with this notation 
Xst^CTst^ Sst.sf 
It is useful to have a verbal rule for writing down such mean products as 
^(SP.,SP.r). 
The following will serve. 
Multiply the detached coefficients of the differentials in the SP's as in ordinary 
multipHcation ; strike out the products in which the related frequencies have no 
common frequency and insert the common part of the frequencies after the related 
coefficients. From the whole subtract the full products of the P's divided by the 
total frequency. This may be proved as follows: 
