a ax" 
Se oe we (2) 
Now it is customary to write the development of ~~ as fol- 
lows, viz. 
—" ~a4ar+ ax? + ax + &e. (3) 
l—a 
and then to commit the mistake of confounding this with the 
series (1) above; overlooking the fact that the ‘‘ &c.” in the one, 
except under particular restrictions as to the value of 2, is 
very different, as to the meaning involved in it, from that in 
the other. 
If we dispense with the ‘‘ &c.” in the series (1), we may 
write that series thus : 
a+tar+az?+ar*+...+4+ az’, (4) 
the sum of which will be truly expressed by the formula (2), 
by making 2 infinite; as that formula is perfectly general. 
But this same formula gives for =. the development 
a ax” * 
[op mat wt ae? + ax +... pax +5 » (9) 
shewing that the “&c.” in (3) differs from that in (1) by a 
quantity which is infinitely great, whenever z is not a proper 
function: except in the single case of x= —1. When z 
is a proper fraction, the two series become identical by the 
x 
ax 
evanescence of ian 
=a 
oe. , : 
It thus appears that [oa iS moe the fraction which gene- 
rates the series (1), x being unrestricted: what this fraction 
really generates is exhibited in (5) above, an equation which 
is always true, whatever arithmetical value we assign to x; 
and to obtain the general expression for the sum of (1), we 
* As the exponent in this last expression is infinite, it seems unnecessary 
to write it @ + 1. 
