142 



th 

 In general if the n fii-st terms of the r power of a multino- 

 mial of n terms derived in this way be equal to the n first 

 terms of the given series (as has been proved to the Case of 



the Trinomial), and the excess of the n + 1 term of the given 



tk th 



series above the ?i+ 1 term of the said r power of the multi- 

 nomial of n terras, be made according to the above method 

 of Extraction to be equal to r times the new member of the 



Root, (this new member to be added to the former multi- 



th 

 nomial). Then, in subducting from the given power the r 



power of the new multinomial, the n first terms are the same 



th th 



as before, and the n+\ term of the r power of the new mul- 



th 

 tinomial will be equal to the n+\ term of the series. For 



— th tk 



both of those terms exceed the ?n- ] term of the r power of 



the former multinomial by r times the term added. Hence, 



in the remainder the ?« + ! first terms will be destroyed, and 



the Index of x will be higher by an Unit in the new member 



of the Hoot than in the preceding member, and therefore 



n-\- I n 



vx will be the «+2 term if ix be the n + l term of the Root 

 (as we have supposed) which will appear by the following 

 General Example, viz. 



GENERAL 



