276 REMARKS ON THE NEGATIVE 
N.B.—These must carefully be distinguished from (sin 2)?, (sin 
2) —1, (sin 2)°, which come under the former or following head, and 
are frequently, though inaccurately, written as above. 
(3). Indices denoting any function whatever,— 
Example (1): Let f1 («) be the differential of « — da, f°? @ 
is a, f— 1 (@) is d~* (a) meaning that which, if differentiated, will 
give z—in other words the integral of z f—? (%)isd~ ? & that 
which, if differentiated twice, will give x, or the second integral of , . 
and so on. 
It will be observed that this illustration shews clearly that a defi- 
nite meaning is attached to the inverse symbol, for although our 
analysis may not be sufficient to enable us, in any special case, to 
integrate the required number of times, yet the operation is not 
only conceivable but never beyond the bounds of possibility, and may 
be practicable, and, what is more, may in every case be performed 
independently of our knowledge of the results of differentiation. 
Example (2): Let f(z) = 2 + = 
f%2 ety 
ean a 
: 2 
For performing the function fi on this we get,— 
a+ J/g —4 D) 2¢2—4 4 2% /72_4 
2 oh i ra 2(@ + / 2A) 
+ 4 
2(¢ + V/72—4) 
=a 
monet 
And similarly, 
pe i= x + wh —4+4 Jf 2(@?—10+ / 7? — 4) 
4 
which may be verified. Beyond this point, the analysis fails to give 
the inverse function, though equations may be found to determine 
them. To take one more example,— 
