tke. Xnji7iiteflmal Calculus, 41 



From each other, or have for their ultimate ration a fatio of 

 >?quality. This being fo, it nluft necefl'arily happen (bccaui'e 

 J^Q, by the firfl; hypothefis, is already fiippofed to approach 

 "continually to MP) that RS will alfo continually approach to 

 the fame line MP ; fo that, like NQ, it will ultimately co- 

 incide with that fame line ; otherwifc it is evident that the 

 ratio of QS to PQ, which, by the fuppofition, ought to ap- 

 proach continually to unity, would recede from it. It is 

 inorcover evident, from the law of continuity, that the fame 

 will be the "cafe with the ratio of RZ to NO, Agreeably, 

 therefore, tb the general notion of differential quantities, 

 above delivered, QtS ought to be the differential of AQ, RZ 

 that of A-Q, QS— jPQ, or NZ-^MO, that of PQ, and 

 laftly, RZ — KOy that plf NO;, and all for the fame reafon 

 that NO', or NQ^MP, is the differential of MP. Accord- 

 ing to t|ie received manner, then, of e.tprfefling differentials 

 in calculation, we muil have QS^dx, RZ^dj, QS — PQ 

 = d{AfO), RZ - NO -r: J(iYO). But we kave already 

 found MO ^ dx, ai?d NO = dyi therefore QS' - PQ = ddx. 

 ^nfJ^^Z-^^Q — ddj', that is, the quantities J<^.r and ddy, (alfo 

 written d\v and dy) will b*i the differentials of the differeii- 

 tials of.r andj^', called alfo, for brcvitv, Jccoyui differences, or 

 differentiats oj^ the fecond order ; that is, ddx is the differentia^ 

 'of the fecond order^ or the fecond difference of x; and ddy 

 that of J.'. 



N0W3 fince QS and PQ are fuppofed to dlifcr Infinitely 

 little from each other, their difference ddx Is infinitely fmall 

 in comparifon with each of them (by article 28). Therefore 

 dil^'ercntials of the febond order are infinitely fmall in cbtn- 

 pariTon with firit differentials, or thofe of the firft order*; 



51. In the fame manner may bie differentia ted, in tticir turn, 



■'• If, inftcad of drawing the new auxiliary line RS in fucH manner 

 that the lines ^ and Pi^dilfer infinitely little from each other, it be drawn 

 in fuch a maiincr that ^^ may be predfcly equal to P^, thit is, fo thit 

 APy ^^, and AS may be in aiithmcticul pro^relfion, wc Ihail have dcix 

 r=:o, or (ix conftant. Thus, one of the d.'flcrentials may be fuppofed 

 conftanti But, from AP /lil^ and AS being in arithmetical progrelfion, it 

 will not follow tljat MP^ N^^^t\d J?.t are lb Jikevvile, unlefs AMN, in- 

 tkad of being a curve, be a ftrai^ht line. Thus, from the fuppofition 

 that ildx = o, it can by no mean* be inferred that tiUv = e. 



' Vol. IX. * F " the 



