ihe- Infinitejirnat Calculus, 45 



Uft iitm dxc^\s\i\t\rm^mi{i\y rriiall m cbmparifon with el^^^H 

 of the others; that is, that the qdotieiit of this laft term b^ 

 either of ihe other teriiis is aii Infinitely frnall qukiitity. r^* 



ther^fof4- 



bf — i or of A' y IS— t.v ^c-, or --5-; 



6f -^,' dt of ir"" ^ is -- iS^ •"* ie; 6r — r- i 



•^r I f —3 • . --3-V*. ci^3-^' 



of — r-5 «r of .V , IS — rv ;r, ot -^-^ 5 



of — , orofA'""^ is— 4Ar i, or -vr-: and fo of others J 



f 4thly, Before^ we cari find the fluxions of powers witU 

 ^(^lional indices, commonly cialled roots, we mull very at-r 

 teiitively cqnfider the law of coritinaatibii in fuch leries as 



A'*, A'^, a:% x^y &c. co'htihually extt-aS:irig the fqnare root, 



I 

 A^, A'% x\ A'^, &;c'. ■ — - — r ^hc cube root, 



Ar'% »v*, x^, A'% &c, ^ — the biquad. root, 



o 3 I i 



x*^, .v% A'^', .v-*", &c. ; — - — — cube root pf the fquarc. 



Here we fee tliat it is not more than certain that x' is the 



il:iuare root of Ar% or x the fquare root of v*, tHclri th^t .i-^ 

 is the fquare root of .r. And, by the fame law that Jt^ is tlie 

 cube root of .v^, and x the cube root of .v*, we may afiirm^ 



that.v^ is the cube rqot of .v, &c. Thus, then, we may fafely 



\^rite X* for V.r, .v'^for ^x^ x-^ for ^x\ See, and univerr 



m 



(ally, Af " for J/x^, This being underdood, it will be evir 

 ^ent that the general formula, itvc "^ x, by writing — for;w, 



m t» — n 



jnuft becomij — « '^ "" ;r, or — x " x. And hence the 



;. • n '' n 



fluxion 



of ^x^ pr ,v % i§ i-r* i = |.v 



t -1 '.i _ 1^ »^ ^ __ 



X 



I « 



^ ^*, (ir #]( is' ft*" '4 = ^h == ^ ; 



2.x' 



of 



