324 Professor Hamitton on Conjugate Functions, 
arenes 
B'=30a +A, 
E =20a + A, 
E =1@a-+a, 
Ko Olay As (100.) 
Boal Sate, 
B= 2a +A, 
B= 3a +4, 
&e. ; J 
in which 
Ora=0; (101.) 
and 
lia = sa, I@pa— Ola, 
Qara+a 2e6a=O0a+Oa, 
3 teat aia gorzerson | | (102.) 
&C. 5 &e. 
The written sign O in Oa is here equivalent to the spoken name zero, as the 
determining ordinal of the null step from a to a, which step was itself also denoted 
before by the same character 0, and is here considered as the zero-multiple of the 
base a; while the written signs 1, 2, 3, &c., in the symbols of the positive multiples 
la, 2a, 3a, &c., correspond to and denote the determining positive ordinals, or the 
spoken names first positive, second positive, third positive, &c.; and, finally, the re- 
maining written signs 1 ©, 2 0, 3 @, &c., which are combined with the written sign of 
the base a, in the symbols of the contra-positive multiples 1 @a, 2 Oa, 30a, &c., cor- 
respond to and denote the determining ordinal names of those contra-positive multiples, 
that is, they correspond to the spoken names, first contra-positive, second contra-posi- 
tive, third contra-positive, &c.: so that the series of signs of multiple steps (99.), is 
formed by combining the symbol of the base a with the following series of ordinal 
symbols, 
conse BESS) Os al a Gs ae (103.) 
We may also conceive this last series of signs as equivalent, not to ordinal names, 
such as the numeral word first, but to cardinal names, such as the numeral word 
one ; or more fully, positive cardinals, contra-positive cardinals, and the null cardinal 
(or number none); namely, the system of all possible answers to the following com- 
plex question: ‘*‘ Have any effective steps (equivalent or opposite to the given 
base a) been made (from the standard moment a), and if any, then How many, and 
In which direction 2”? In this view, 3 © is a written sign of the cardinal name or 
