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tion (1), without any further assumption as to the nature of 
the operations denoted by 7 and p, we may change 7 into p, 
and p into —7; so as at once to form the correlative equation, 
¢ (ep, —7) = 9; (4) 
for this latter will be deducible from the primitive, in the form 
(2), by the same processes, whatever they are, which conduct 
us from (1) to (3). 
The value of this principle must depend upon the extent 
of its application; and this will be found much wider than 
might at first sight be supposed. Forasymbolical equation of 
the form (3), which is verified for any subject whatever ope- 
rated on by its left-hand member, if it be not, in its existing 
state, identically true, must hold good in consequence of our 
being able to transform it into one that is identically true by 
means of the fundamental equation (1), which connects 7 and 
p- ‘Thus we may regard all useful equations of the form (3) 
as deductions from the single primitive (1). 
Of the general nature of the results which may be deduced 
from this one very simple equation, and that without the in- 
troduction of any fresh hypothesis as to the operation of 7 and 
p, the following example will give a sufficient idea. 
Making a= 1, which does not much diminish the gene- 
rality of our conclusions, we have 
pr=7p+1, 
pm’ = 1pt + 7, 
=7(zp+1)+7; 
=7’p +27. 
Again, 
pt = mom + 27”, 
=m" (rp +1) + 27’, 
=mp+3r’. 
And for x any positive integer we get, 
pm" = 1"p + nmr"-*. 
