147 



But according to the algebraical disposition of the terms, 



5 



l + t'-^ would be thus expressed 1+5 x -'--f-lOx -i-+iOx 73-^^+ 

 5 X T-- — + — ' . 



LOOOO ' lOOOOO 



The general Example will apply to the latter disposition, 

 but if we use the former arrangement the Extraction of the 

 Square Root will depend upon trial. 



CoR. T. We have hitherto considered the Indices of x in 



the r Root of l+x, or in \+x; but if the Binomial is />+x, 

 by resolving it into jjx (H-^, the same proof will extend to 

 the Law of the Indices of the second member; and therefore 



{l+p)~-^+?^"*'^P^+&c. Ilcnce we shall have p+x =p x (l+-) 



=P+-r>'P + cxp^+Scc.+txp'--^kc.=p-t'~xp-{-cxp+&cc.+txp+Scc. 

 In the last step of this proof I have supposed the equality 



pr T " 



of ^^ and /., which perhaps may require a Demonstration when 

 ^ is a Fraction. If the Indices oi p in the Dividend and 

 Divisor were Integers, it would follow from notation that the 

 subduction of the Index of the Divisor from that of the Di- 

 vidend should denote a Division; viz. if »i and nr be Integers, 



;," 



we shall havef^r:^; and hence we shall prove that the 



subduction of the Index of p in the Divisor from the Index 



of p in the Dividend will eifect a Division, although p should 



^°^- ^^i- X bave 



