[ 354 ] 



the fluxions, making x^y, &c. conftant, it follows that z may 

 be deduced from z by taking the fucceflive fluxions of z by the 

 former problems. 



" • 77. 71 — I 1"+' x" 



Examples. The fluent of x A; = Cor. -{ ■—— x~ hCor. 



1.2 /2+I 7l-\- I 



The fluent of 3 x-y x ■\- x' y ■\- 1 xy' x + i x^ y y — (taking x 

 and y — 0, and fubftituting for x and y, x and y) 

 3. 3. 2 xiy + 3. 2 x^ y + 2. 3. 2 x^ y- + 2- 3- 2 x'jy' _ ,,.. 

 I. 2. 3. 4 



The fourth example when n is odd is an inftance of finding 

 fluents a priori hy this method. If x = Yj, where Y is an algebraic 

 funflion of j', then by common algebra reducing this equation to 

 integral values, and taking the fluxions particularly by the former 

 rules, it will be known whether x the fluent can be had in finite 

 terms; in fome cafes, very readily, in many, however, the difiiculty 

 will greatly exceed the inverfe method, but this difficulty may be 

 probably obviated by given the fubjed that attention it feems to 

 deferve. 



But it ought to be remarked when theie are two or more inde- 

 pendent variable quantities, that the given fluxion muft be poflible, 

 that is, muft have originated from a fluent. Thus for inftance 

 J ;f is not a poflible fluxion, for it cannot have originated from 

 any flowing quantity wherein x and y are independent. 



Th? 



