140 

 Demonstration. The first member of the power behig 



r 



Unit, the Subduction of 1 and the division of the first term of 

 the remainder by r gives ^x for the second member of the 



Root; by subducting again the r power of the Binomial quan- 

 tity 1 + "^ from the given power, the two first terms of the 

 power are destroyed, for by the above method of extraction 

 of Roots, tlie second member of the Binomial was derived 



by dividing the second member of the series by r, and this 



th 

 is again multiplied by r in the second member of the r power 



of the Binomial found, therefore the first and second 



th 

 terms of the r power of the Binomial found being sub- 

 ducted respectively from the first and second terms of the 

 given series, the first dimension of x will not appear in 



the remainder, and therefore the next term is of two di- 



th 

 mensions. The two first terms of the r power of the Tri- 

 nomial are the same as those of that of the Binomial found. 

 But from the above nature of Involution, the third term of 



th 

 the r power of the Trinomial is greater than the third term of 



th 

 the r power of the Binomial by an excess which is the third 



term of the Root multiplied by r, and from the above nature 

 of extraction of Roots, the third term of the given series is 



th 

 also greater than the third term of the r power of the Bino- 

 mial found, by an excess which is also the third term of the 



Root 



