274 REMARKS ON THE NEGATIVE 
by what is convenient than by what is true, and that perhaps some 
other more convenient explanation might some day replace the one 
now adopted. But in the case of the negative index such a mode 
of expression is still less admissible, because the steps by which the 
meaning is established are so easy and straightforward. 
If any operation performed on a quantity x be denoted by f? (x), 
we should denote the same operation performed upon yf 1 (#) by 
i ( Vp 1 (2) ) or conveniently by St 3 (a). Pe (x), therefore, denotes 
the operation f 1 performed once upon Wii 1 (x), or twice successively 
on «. Similarly f° (x) may be used to denote the function uh 1 per- 
formed once on I 2 (x), twice successively on em (x), or three times 
successively on 2, and soon. Adopting this notation we shall have 
yh m (x) to represent the operation ie 1 performed m times on # suc- 
cessively, and j™ +” (x) or Wi n +m (x) to represent either the per- 
formance of the operation ue 1 m times on Sf ” (a); ¢.e., == I m ( Sj (x) ) 
or » times on f'™ (z) =a ife ( dh uN) ) orm + n times on 2, the 
result being in each case the same, 7.e., 
[ere O) Sj (Gi 2 @) ) (a) 
=fn (fa (2%) @) 
Hence f” (2) is derivable from hi +” (x) by undoing the n opera- 
tions denoted by f” in (a) and f™ (x) = fm +"—” (2). 
Hence — n in the index must be regarded as undoing the opera- 
tion f 1 n times supposing it had been performed more than n times 
on @. 
But what does f° (a) or f—* (x) represent of itself, when there 
is no operation to. undo P 
Now we observe that i. 1 denotes an operation performed once, 
fie twice; f™ m times. 
5) Wh represents the operation performed zo times, that is, not 
performed at all, or t © (@) is the same as 2, for just as truly as ie 
represents m operations, so truly does St ° represent no operations : 
