40 Carmt 9n the Theory tf 



therefore, dxdy be neglected in the preceding equation, Jt 

 Nvill then become d,xy— xdy -^ydx^wlxich is what I call an 

 iniperfetSl equation.^ But, fincc imperfect equations (by ar- 

 ticles 31 and 34) may be employed like rigorous ones, with- 

 out inducing any error into the refult, it is evident that I may 

 life this lart equation inflead of the firft; and, 4s it is more 

 ^mple, I (liall by its help abridge my calculation. 



I fay, then, that the difterential of the produ6l of two va- 

 riable quantities, is equal to the produ(!^ of the firfl variable 

 quantity into the differential of the fecond, phis the product 

 of the fecond variable quantity into the differential of the 

 firft. And this propofition will be one of thofe which (ift 

 article 35) I have called imperfcd propo(?tions, that Is, which 

 ;ire capable of being exprefTed by impcrfcvt equations, and 



of ^^, or Af ^ IS \x^ X = \x X == — J 5 



4^^ 



of V x^-. or A.'^, is \x 



tjs 



f-l. , •^. 2^' 



X = 



and fo in fimilar cafes. . - 



5thlv, The fluxion of a fraction maybe fcund by confider^ 

 ing it as the producSl of the numerator 'and denominator, 



giving the latter a negative index, Xhu^ — is equivalent to 



y X X \ and as the fluxion of x-^ is y^ + xy^ fo- the fluxion 

 X . — I 



pf -^ , or of its equivalent y X x muft be, 



■ ' isr — ■ "~ 



(r xi)+(-rix.v) 





The fluxion of a fraftlon, — _, may alfo be found, but not 



y 



fo elegantly, by aftually dividing ^ 4- x by y -^ y. 



The above is a fpccimen of a very eafy, and, in fome mea-: 

 fare, new, manner of treating the fundamental proccifes of 

 tiuxions, which I long ago mentioned to ISIr. Tilloch, and 

 which, with fome thoughts on prime and ultimate ratios, 

 J intend to offer him for publication, when I have time to 

 ^raw up the paper, and he has room to infert it in the Philo- 

 fophical Macrazine,— .W. D. 



• ^ which. 



