THE 



1 



If a is a constant, we have 



i 



(118.) Aatpz = a(<pz -^ Ayz) — atpz = aAyz — (a A<px) y ayx , 



A 2 a(px = — Aa(px,aAz, etc. 



A((px)a=(A(pz)a — ((A(pz)a).(pxa, 



A 2 (<pz)a = — A((pz)a, etc 

 (119.) A(«,9pj) = a.Ay* . 



Let us differentiate the successive powers of x. We have to ihe first j ce, 



A(x 2 ) = (z + Az)z—z 2 = 2Jr«- tI ,(A*) t » + (A#)», 



Here, if we suppose A# to be relative to only one individual, (Ax) 2 vanish* and 



we have, with the aid of (115), 



d(x 2 ) = 2.X l y (Jx . 



Considering next the third power, we have, for the first difi ntia). 



A(x3) = (x-\- Az)3 — x3 = 3 .x3-<t,(Axy> + 3 .x3- t2 ,(Axr 2 + (Ax)3 , 



d(x3) = 3 .z 2 ,(dx) . 

 To obtain the second differential, we proceed as follows : 



A2-(s3) = (x + 1. Az)l — 2.{Z + Az)3 + X3 



= x3 + 6.z3~'\{Azy* + I2.*3-*,(A*)* + %.(Az)3 



2.z3 — 6.z3-\(Azft — 6.*3-*»,(A*)h - a .(A* 



-\-z3 





\ 



6.z3- t2 ,(Axf 2 + 6.(Ax)3. 



Here, if Ax is relative to less than two individuals, A<px vanishes. Making it relatn e 



7 



to two only, then, we have 



d}-(z3) = e.z\(dzy. 



These examples suffice to show what the differentials of # will be. If for the number 

 It we substitute the logical term n, we have 



AM = (s + Axf - * = [«].^- t, ,(A*) tI + etc. 



d(z») = \ri\.x n - l ,(dx). 



