30 



Now it is customary to write the development of as fol- 

 lows, viz. 



■r— — = a + ax -I- ax^ + ax^ -f- &c. (3) 



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 a;, 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 : 



« 4- «!^ + 0,0^ + «*^ 4- • • • + «^°°j (4) 



the sum of which will be truly expressed by the formula (2), 

 by making n infinite; as that formula is perfectly general. 



But this same formula gives for the development 



= a + «a? + ax^ + ax^ + . . . + ax^ 4- , (5) 



\—x \—x 



shewing that the " &c." in (3) differs from that in (1) by a 



quantity which is infinitely great, whenever x is not a proper 



function : except in the single case of a; = — 1 . When x 



is a proper fraction, the two series become identical by the 



„ ax'' 



evanescence oi :; . 



\ — x 



It thus appears that is not 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 oo + 1 • 



