INDEX OF A FUNCTION. 275 
one is as general as the other. Hence we can assign the meaning 
of the negative index, for af —1 means the reverse operation to I 1. 
if both iA 1 and vi —1! be performed on 2, one undoes what the other 
does, and the result is 2. So that if — 1 () represents that quantity 
or quantities, for there may be more than one, on which if you per- 
form the functicn denoted by up 1 the result is 2. And so Fe — 2 de- 
notes that quantity or quantities on which if you perform the fune- 
tion f } twice the result is x. It will not be possible in every case 
to assign a numerical or even symbolical expression of every inverse 
function that may occur, but it appears to me that the meaning of 
the notation is perfectly definite, and that it ought to be treated as 
such. The theory of indices stands on very different grounds from 
any arbitrary convenient explanation of, for instance, the symbol 
/ —1, derived from the truth of results obtained by treating it as a 
real quantity. It may, however, be as wel! in conclusion to notice 
one or two obvious cases to which the above remarks are applicable: 
(1). Theory of Indices in multiplication or division of like quan- 
tities in arithmetical algebra,— 
Here a” —-a X @ X a to m factors. : 
Now a denotes an operation performed on unity, namely, multi- 
plying it by a. Hence a replaces f1 and 1 replaces z, 1 being 
usually for simplicity omitted. Thus a = a@° (1)= 1. 
a —} =a quantity which, multiplied by a, will = 1, z.e.. = 
a@—? —a quantity which, multiplicd twice by a, will give 1, 2.e., 
al= 
4 
1 
q2, and so on. 
Unity is here abstract or concrete, and the result abstract or con- 
erete accordingly. In the few cases in which an interpretation may 
with more or less strictness be applied to the multiplication or divi- 
sion by one another of concrete magnitudes, the unit will of course 
be of that denomination which is denoted by the index after such 
multiplication or division. 
(2). Indices denoting Trigonometrical Functions, for example,— 
Sin © (x) means 2. 
Sin (2) «the sine of 2. 
Sin-1 (2) “ that angle of which the sine is <. 
Sin 2? (a) “ the sine of the sine of a, and 80 on. 
