133 



the Index in a Fractional Forn>, and where the Index w-its- 

 leally a Fraction, he applied with success the same form 

 which aheady had served when the power had an Integral 

 exponent. That this Induction however is not a legitimate 

 proof, will appear from distinguishing the particular cases 

 which are comprehended under this general mode of no- 

 tation. 



Wlien the Index of a power of a Binomial is an affirmative 

 whole number, that power is produced by repeatedly multiply- 

 ing the Binomial quantity by itself so often as shall make the 

 number of multiplications to be less by one than are the units 

 in the Index. of the power: and it follows from the arrange- 

 ments of common multiplication, that the power of a Bino- 

 mial quantity whose Index is the whole number m, as i +aV 



I 3 4 



will have the following Form, viz. l-\-mx + Cx-\-T>xi- Ea;' + &c. 



m 



+ Zx. The Index of such a power, as well as every other quan- 

 tity in numbers, can be represented after the manner of a 



Fraction as rtiT—' But, if the Index is really a Fraction, the 



power cannot have arisen as above from a continued multi- 

 plication of the Binomial quantity by itself, since a mullipli- 

 cation can no more be repeated a Fractional than a negative 

 number of times. The analogy, tlierefore, which is founded 



on the consideration of 1+*' under the above notion of 

 powers, will be insufHcient to determtne even the Form of a 



T 2 power 



