312 Professor Hamitton on Conjugate Functions, 
and the theorem of inversion (12.) may be written thus : 
0 b=Oa, if b=a, (52.) 
The corresponding rules for inverting a non-analogy shew that, in general, 
© +02, if ba; (53.) 
and more particularly, that 
Ol '<Os, iPS a (54. 
and Ob>@a, if b< a, (55.) 
It is evident also that , 
if Y=@a, then a=0 #; (56.) 
that is, the opposite of the opposite of any proposed relation ais that proposed 
relation itself; a theorem which may be concisely expressed as follows : 
8 (8 a) —=2)5 (57.) 
for, as a general rule of notation, when a complex symbol (as here © @) is substituted 
in any written sentence (such as here the sentence »=© 2’) instead of a simple 
symbol (which the symbol «', notwithstanding its accent, is here considered to be), 
it is expedient, and in most cases necessary, for distinctness, to record and mark this 
using of a complex as a simple symbol, by some such written warning as the enclosing 
of the complex symbol in parentheses, or in brackets, or the drawing of a bar across 
it. However, in the present case, no confusion would be likely to ensue from the 
omission of such a warning ; and we might write at pleasure 
9(Oa)=a, 8 fOa=a, Of[Oa] =a, O Oa= a, orsimplyO@O2=8. (58.) 
10. For the purpose of expressing, in a somewhat similar notation, the properties 
of alternations and combinations of analogies, set forth in the foregoing articles, with 
some other connected results, and generally for the illustration and development of 
the conception of ordinal relations between moments, it is advantageous to intro- 
duce that other connected conception, already alluded to, of steps in the progression 
of time ; and to establish this other symbolic definition, or conventional manner of 
writing, namely, 
B=(sp—A)+A4, or B=a+a if B—A=ajz (59.) 
this notation 2+a, or (B—A)+a, corresponding to the above-mentioned concep- 
tion of a certain mental step or act of transition, which is determined in direction 
and degree by the ordinal relation a or gp—a, and may, therefore, be called ‘the 
