44 CartiGt on the Theory of 



Biit, with rcfpe^t to this equation, I obfenc^ that dx and 

 4y, being uifmitcly fmall, in coinparifon with x and jy, the 



hi\ 



But, for a reafon which the author gives, xy (in his notatioii 

 axJy) may be rcje^Slea, and ib the fluxion 6t\i;y is ,\J -\- yx, 



kxample 2. Thus alfo, the fluxion of 



■• xyz is x-zy 4- y'zx -t- .rnxi. 



Exdmple 3|. In Hke manner, the fluxion of 



vx\-^, is vy^ 4- vx7J( + v^2i' -f- ^«^'v.. ,.,i„ 

 ,, 2aly, To find the fluxions of pow^cxs, as ;k%,^V<^F- - . 



Example I. In the firft example 2^bove, we Had ^ -f-' J'^ /of 

 the fluxion of xy. Let .v -- )', then xy — .r", ah(i xy H- jf 

 = A\r + A'i or aA'i, will be tne fluxion of ,r*. 



Example 2. In the fecond example above^ let x -- y = t 

 then ajis will become x^, and a;^ + vzf -{• y^x willbcrome 

 XXX 4- .v.ii -\- Avvi-, or ajr ,*•, \vhich will be the fluxion of .v\ 



Example ■^. In like manner, if^ in the ^d example abovCj^ 

 v rz X — y — Zj the fluxion of v* will be l^x^x, , 



Tii'Jli the flux'ioli of x'^ being clxx, that of .r^ teiftg ^v^.r, 

 that of .f* being 4t\v^ &:c. we conclade that the fluxion or 



any power qf^.v \vhatever, fay x , will be inx x. Here 

 the index m is fiippofed tq be integral and ^flirmative ; btif 

 the fame forniula, 'mutatts viutand'is, ierves where rn is frac- 

 tional or negative. 



3dly, Before we proceed to powers whofe indices are ne- 

 gative, it will be iiecefl'ary to contemplate very attentively, 

 iuch fprics as 



x^,_ x\ ~x', x', x-\ ---, --r , — 3-> -^3 ^:<^. 



.r- ■ X- X' .V 



or, .r4, x\ x\ .v% .i'\ X , a; y x , x , &lc. 

 i>rhere the law of continuation of the. pqwprs is a fuccell;ve 

 diviiion by Xy and that of the indices a Uicceilivc fuDtraiSlibn 



of I ; fo that x"^ j .v , &c. in the fecond feries^ is equiwi-. 



lent to — -, -X5 <^G' '^^ the firfl;; or, in other woris, thtj 



Jt X 



— r —2 

 i^ogative indices — I, — 2> &:c. of a- ", * , &^c. con- 

 fidered as multipliers, are equivalent to the pofltive iiidices 



I, 2, &c. of -^, — 7", &c. co.nfideripg .1% .r% &c. as di- 



VI ^ r 



vifqrs. lieace thp general formula, mx x, becomes 



-^anx X, OT — — — ; and thus the fluxion 



©J 



