392 Professor Hamitton on Conjugate Functions, 
the prefixing of the mark + to an isolated symbol of a step or of a number, does 
not change the meaning of the symbol; but the prefixing of the mark — converts 
that symbol into another, which denotes the opposite of the original step, or the 
opposite of the original number ; so that the series of whole numbers (103.) or (266.) 
may be written as follows : 
Q 
—3, —2, —1, 0, +1, +2, +3, ... (364.) 
Also, in this notation, 
bt(+a)=bta, D+(—a)=bFa, (365.) 
b+(t+ta)=b+a, b+(-a=bFa. . 
36. Finally, as we wrote, for the case of commensurable steps, 
Vv aay, 
a aay 7c 
» and vy being here whole numbers, so we may agree to write, in general, 
2B Bed eek (366.) 
eee AG 
whatever ratios a and 6 may be; and then this symbol » will denote, in general, the 
algebraic quotient obtained by dividing the number or ratio } by the number or ratio 
a; whereas we had before no general way of denoting such a quotient, except by the 
mark u prefixed to the symbol of the divisor a, so as to form a symbol of the reci- 
procal number u a, to multiply by which latter number is equivalent to dividing by 
the former. Comparing the two notations, we have the formula, 
7 = 144, (367.) 
and generally 
b 
SSG <i SS Oe Ue (368.) 
a 
These two marks © and u haye been the only new marks introduced in this Ele- 
mentary Essay ; although the notation employed for powers differs a little from the 
common notation: especially the symbol Be; suggested by those researches of Mr. 
Graves respecting the general expression of powers and logarithms, which were the 
first occasion of the conception of that Theory of Conjugate Functions to which we 
now proceed. 
END OF THE PRELIMINARY AND ELEMENTARY ESSAY. 
