390 Professor Hamitton on Conjugate Functions, 
step to the moment, or the composition of the two steps with each other. For the 
decomposition of a step into others, we have used no special mark; but employed 
the theorem that such decomposition can be performed by compounding with the 
given compound step the opposites of the given component steps, and a special nota- 
tion for such opposite steps, namely, the mark* © prefixed; so that we have written 
© a to denote the step opposite to the step a, and consequently 6a + b to denote the 
algebraical excess of the step b over the step a, because this excess may be conceived 
as a step compounded of b and Oa. However, we might have agreed to write 
(») +4)—(ata)=b—a, (354:.) 
denoting the step from the moment a+a to the moment b+ a, for conciseness by 
b — a; and then b —a would have been another symbol for the algebraical excess of 
the step b over the step a, and we should have had the equation 
b—a=Oadb. (355.) 
We might thus have been led to interpose the mark — between the marks of a com- 
pound step b and a component step a, in order to denote the other component step, 
or the algebraical remainder which results from the algebraical subtraction of the 
component from the compound. 
Again, we have used the Greek letters » v & p w, with or without accents, to denote 
integer numbers in general, and the italic letters ik 1mm to denote positive whole 
numbers in particular ; using also the earlier letters a B y aed to denote any 
ratios whatever, commensurable or incommensurable, and in one recent investigation 
the capital letter B to denote any positive ratio: and employed, in the combi- 
nation of these symbols of numbers, or of ratios, the same marks of addition and of 
opposition, + and ©, which had been already employed for steps, and the mark of 
multiplication x , without any special mark for subtraction, We might, however, 
have agreed to write, in general, 
(6 x a)—(a x a) =(b—a) xK ay (356.) 
as we wrote 
(b x a)+(a x a)=(b+a) x a; 
and then the symbol 4—a would have denoted the algebraical excess of the number 
* This mark has been printed, for want of the proper type, like a Greek Theta in this Essay: it was 
designed to be printed thus G, 
