148 
Since p is distributive, we shall have 
p Sp=Jp ps 
and adding these equations together, we get 
p(w +Jp) = (a+ fp) p+], (12) 
an equation still of the same form as (1). And, therefore, in 
any symbolical equation deduced from (1) merely in virtue 
of its form, we are at liberty to change 7 into 7+fp. Similar 
reasoning will show that in symbolical formule obtained in 
the same way, we may change p into p+fz. As particular 
cases of this we may observe that in any symbolical equation 
involving z and D, we are at liberty to change z into 2+/D, 
or D into D+ fe. 
Again, if we operate on (5) with (fz), it becomes 
(Uap dam da (Yr) p 41, 
inasmuch as any two functions of 7 are commutative. Now 
this again is an equation of the form 
pr=7ptl, 
(Wr) p being put for p, and ym for z. 
It follows then that in any deduction from (1) we may 
change 
p into (pr) "p, 
and (13) 
w into Wr. 
In like manner, if we operate on (7) with (¢’p)', it be- 
comes 
gp (pp) *7=(~'p)*rop +1; 
showing that in any deduction from (1) we may change 
p into ¢p, 
and (14) 
m into (¢'p)' 7. 
Writing « for 7, and D for p, we learn from (13) that it is 
legitimate in symbolical formule to change z into yz, and D - 
