149 



Cor. III. We have now deduced the Law of the Indices 

 of the terms from the extraction of Roots, and from hence 

 also the Coefficients of the second and third terms where the 

 Index of the power is a Fraction will be known from the 

 cases of Involution, wherein the Index of the power being 

 an affirmative Integer, the dependence of the Coefficient on 

 the Index is known. 



Thus rb—m therefore ^:r" and in negative powers — "=~y 



Also rc=C — c— "'—"' __'•— r^ I therefore c= 



1.2"~1.2 J.2.r» 



and in negative powers "* — p— 2m— m+wr _ m+mr 



Having now proved that p+x ' will be of the same form, 

 whether the Index is affirmative or negative, I shall hereafter, 

 for the sake of convenience, put " instead of +_y, and then 

 the general Index " will denote any positive or negative 

 Fraction or whole Number. 



If 2 is a surd, we can find rational numbers which will 

 approach it as near we please, and although we cannot con- 



m 

 T 



elude from the rules which I have given that /'+i' will be of 

 the above form, when " is a surd, since no such arithmetical 

 process as Involution, Evolution, or Division, can in that 

 ease be understood, yet the above forms will apply to the 

 powers which have the approximate value of " for an Index. 



X 2 CHAPs. 



