and on Algebra as the Science of Pure Time. 421 
And the general inverse exponential or logarithmic function-couple, which may, by 
(129.), be thus denoted, 
(2, r,)=0' CY Yr), if (YW Y2) =P (X,, 22), (179.) 
may also, by (174.) and (176.), be thus expressed : 
ig =F Yom), 
® CY ==, b)? (180.) 
it involves, therefore, two arbitrary integer numbers, when only the couple (¥,, y;) 
and the base (b,, b,) are given, and it may be thus more fully written, 
wo" a ee Cy ’ Yo) 
®"'(Yis Yo) = log wy my + (Yo Y2) =F : 
Fa( b,, b,) (181.) 
For example, the general expression for the logarithms of the primary unit (1, 0) to 
the base (e, 0), is 
(0, 20’ z) (2 w' 7.0) 
ea ely er OE Gira mea co aay 3 
lo} 
wo 
(182.) 
or, if we choose to introduce the symbol “—1, as explained in the 13th article, 
that is, as denoting the couple (0, 1) according to the law of mixture of numbers with 
number-couples, then 
2 war JT 20 7 
OS e* Ser Li eee (183.) 
In general, 
nf EY Yo) + (0, Qo! a 7 
198 orm)» (I W=EI CH, In) + 0, 20) oe 
The integer number w may be called the first specific ordinal, or simply the orpEr, 
and the other integer number w’ may be called the second specific ordinal, or simply 
the rank, of the particular logarithmic function, or logarithm-couple, which is deter- 
mined by these two integer numbers. ‘This existence of two arbitrary and inde- 
pendent integers in the general expression of a logarithm, was discovered in the year 
1826, by Mr. Graves, who published a Memoir upon the subject in the Philosophical 
Transactions for 1829, and has since made another communication upon the same 
-subject to the British Association for the Advancement of Science, during the meeting 
of that Association at Edinburgh, in 1834: and it was he who propesed these names 
of Orders and Ranks of Logarithms. But because Mr. Graves employed, in his 
