the Infinitejiiruil C(JlIcuIu5, 45 



idiflTerentlal quantities by procefles which lliow the method of 

 returning from a differential to its integral. 



Mj; prefent objc6l is 7iot to write a ireatije on thcfe caUuViy 

 hut only to give ihefundairiental rules, and tojhoiu that theft 

 rules are only fo ma?iy applications of the general principles 

 which have been explained, 



54. Let it firlt be propofed, thcn^ to fiffign the differential 

 of the fum, x + jv 4- « &c. of feveral variable quantities. 



By the hypothefis, x becomes x 4- dx^ y becomes y -{■ dy 

 &C. Therefore the fum propofed becomes a; 4- dx ^y •{- dy 

 4- 2J + ^s? 8cc. Confequently, it is inereafed by dx •{- dy + 

 dz. Sec; and thefc iacrements are precifely what we have 

 called differentials *. 



55. The differential of fz 4- ^ 4- ^ 3tc. 4- .v 4- j/ 4- zhc» 

 is now required; a,hfC, 8cc. being conftant, and x^yyZf^c, 

 variable^ <^uantities. 



By the hypothefis a remains a, h remains h, Sec. and Jt 

 becomes !<'\-dx, y beconues y 4- dy he. Therefore the fum 

 propofed becoi>ies a 4- ^ + ^ &c. + x •\- dx &c. Confe- 

 quently it is increafed by dx -\- dy + dz &c,; and this incre*. 

 rucnt is the differential fought, which is the fiime as if there 

 iiad been no conftant quantities in the propofed fum. 



Required the differential of ^.v. 



By the hypothefis^ the conftant quantity a remains un- 

 changed, and the variable quantity x becomes x + dx; there- 

 fore ax becomes ax + adx ; and confequently <m? is increafpd 

 by adx, which is the differential fought. 



56. Required the differential of xy. 



From what ha!s been faid, it appears that the differential 

 here required is ydx + xdy 4- dxdy, that is, we have 

 d , xy = ydx 4- xdy 4- dxdyf. 



^ ' ^ But, 



* -See the nole at the end of article 49, 



t As the ingenious author has touched no farther on the 

 practice than feemed necellary to elucidate his theory, I (liall 

 endeavour to fliow (as plainly as I can in a note) how to find 

 the differentials, or, which is the fame thing in pradice, the 

 fluxions of products, powers, roots, and fractions. 



ifi. To find the fluxions of proiudts, fuch as a;', xyz, Sec, 



/^xampU I. {x + x) X ( v +» = .^V 4- xy 4- yx + ^y- 



F ^ But, 



