148 
Since p is distributive, we shall have 
p Se =Jp p> 
and adding these equations together, we get 
p(w +Jp)=(7+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 r+fp. Similar 
reasoning will show that in symbolical formule obtained in 
the same way, we may change p into p+/fm. As particular 
cases of this we may observe that in any symbolical equation 
involving # and D, we are at liberty to change « into #+fD, 
or D into D+ fe. 
Again, if we operate on (5) with (f’7)-1, it becomes 
(Wr) pdr=da (Yr) *p +1, 
inasmuch as any two functions of 7 are commutative. Now 
this again is an equation of the form 
pr =7p+1, 
(Wr) 1p being put for p, and pr for 7. 
Tt follows then that in any deduction from (1) we may 
change 
p into (p'r) *p, 
and (13) 
w into dr. 
In like manner, if we operate on (7) with (¢’p)}, it be- 
comes 
ge (pp) *7=(9'p)*rHp +1; 
showing that in any deduction from (1) we may change 
p into ¢p, 
and (14) 
m into (¢'p) 7. 
Writing 2 for 7, and D for p, we learn from (13) that it is 
legitimate in symbolical formulz to change z into Ya, and D - 
