FLUXIONS. 



yvxz-\-zvxy. Again, the fluxion of 

 a -f- x X b y = a b-\-bx ay x> y, , 

 is b x a y x y y x. 



4. To find the fluxion of a fraction, the 

 rule is, from the fluxion of the numerator, 

 multiplied by the denominator, subtract 

 the fluxion of the denominator multiplied 

 by the numerator,and divide the remainder 

 by the square of the denominator. Thus, 



the fluxion of- is 



* 4- 



so of others. 



In the examples hitherto given, each is 

 resolved by its own particular rule ; but 

 in those that follow, the use of two or 

 more of the above rules is requisite: thus 

 (by rule 2 and 3) the fluxion of x* y 1 is 

 found to be 2 x 1 y y -f 2 y- x x j that of 

 x 1 



, is found (by rule 2 and 4) to be 



2y*xx 2/r 1 */w ..'.. r-x* V z 



- --- *. and that of , is 



(by rule 2, 3, and 4,) found to be 



z x* y* z. 



5. When the proposed quantity is affect- 

 ed by a coefficient, or constant multiplica- 

 tor, the fluxion found as above must be 

 multiplied by that coefficient or multipli- 

 cator: thus- the fluxion of 5 xi, is 15 x* x; 

 for the fluxion of xi is 3 x 1 x, which mul- 

 tiplied by 5, gives 15 x 1 x. And, in the 

 very same manner, the fluxion of a xn will 

 be n axn 1 x. 



Hence it appears, that whether the root 

 be a simple or a compound quantity, the 

 fluxion of any power of it is found by the 

 following general Rule : 



Multiply by the index, diminish the in- 

 dex by unity, and multiply by the fluxion 

 of the root. Thus the fluxion of z 8 = 

 8 zi z : the fluxion of 4 xs = 24 x$ x and 



3 12 



the fluxion of-r = OQ z 7- * = 



3z 



of this kind to enter upon the second, 

 third, &c. fluxions, we shall therefore 

 proceed to 



FLUXIOJTS, inverse method of, or the 

 manner of determining the fluents of 

 given fluxions. 



If what is already delivered, concerning 

 the direct method, be duly considered, 

 there will be no great difficulty in con- 

 ceiving the reasons of the inverse method; 

 though the difficulties that occur in this 

 last part, upon another account, are in- 

 deed vastly great. It is an easy matter, 

 or not impossible at most, to find the 

 fluxion of any flowing quantity whatever; 

 but, in the inverse method, the case is 

 quite otherwise; for, as there is no me- 

 thod for deducing the fluent from the 

 fluxion a priori, by a direct investigation, 

 so it is impossible to lay down rules for 

 any other forms of fluxions than those 

 particular ones, that we know, from the 

 direct method, belong to such and such 

 kinds of flowing quantities ; thus, for ex- 

 ample, the fluent of 2 x x is known to be 

 x 1 ; because, by the direct method, the 

 fluxion of a; 1 is found to be 2 x x: but the 

 fluent of y x is unknown, since no expres- 

 sion has been discovered that produces 

 y x for its fluxion. Be this as it will, the 

 following rules are those used by the best 

 mathematicians, for finding the fluents of 

 given fluxions. 



1. To find the fluent of any simple 

 fluxion, you need only write the letters 

 without the dots over them: thus, the 

 fluent of x is x, and that of a x -f- b y, is 

 a x + b y. 



2. To assign the fluent of any power of 

 a variable quantity, multiplied by the 

 fluxion of the root ; first divide by the 

 fluxion of the root, add unity to the ex- 

 ponent of the power, and divide by the 

 exponent so increased : for dividing the 

 fluxion n x ni by x, it becomes n xni- 

 and adding 1 to the exponent (n 1) we 

 have n x n which divided by n, gives x" 9 

 t\e true fluent of n xni x. Hence, by the 

 same rule, the iuent of 3 x 1 x will be 



= xi; tiatof2x5a: =^ ; that of y $ y 

 = 1 y 3 - ; thai^f a y ^y = 3 a y~S; and 



Having explained the manner of de- 

 termining the first fluxions of variable 

 quantities, it is unnecessary in a work 



that of y 



y - 



