FLUXIONS. 



I 



Thus, let the point m be conceived to 

 A m m r 



1 j 



move from A, and generate the v..riaWe 

 right line Am, by a motion any how regu- 

 lated; and let its velocity, when it arrives 

 at any proposed position or point R, be 

 such as would, was it to continue uniform 

 from that point, be sufficient to describe 

 the line Rr, in the given time allotted 

 for the fluxion, then will Rr be the flux- 

 ion of the variable line A m, in the term 

 or point R. 



The fluxion of a plain surface is con- 

 ceived in like manner, by supposing 1 a 

 given right line mn (I 'late V. Miscel. fig 1 . 

 8) to move parallel to itself, in the plane 

 of the parallel and moveable lines AF and 

 BG : for if, as above, R-r be taken to ex- 

 press the fluxion of the line A w, and the 

 rectangle R r s S be completed; then that 

 rectangle, being the space which would 

 be uniformly described by the generating 

 line m n, in the time that A in would be 

 uniformly increased by m r, is therefore 

 the fluxion of the generated rectangle 

 B m, in that position. 



If the length of the generating line mn 

 continually varies, the fluxion of the area 

 will still be expounded by a rectangle un- 

 <ler that line, and the fluxion of the ab- 

 sciss or base : for let the curvilinear 

 space A n m (fig. 9,) be generated by the 

 continual and parallel motion of the va- 

 riable line m n ; and let R r be the fluxion 

 of the base or absciss A ni } as before, then 

 the rectangle R r s S will be the fluxion of 

 the generated space A m n. Because, 

 if the length and velocity of the gene- 

 rating line m n were to continue invaria- 

 ble from the position R S, the rectangle 

 R r s S would then be uniformly gene- 

 rated with the very velocity wherewith 

 it begins to be generated, or with which 

 the space A m n is increased in that posi- 

 tion. 



FLUXIONS, notaiion of, of invariable 

 quantities, or those which neit'her increase 

 nor decrease, are represented by the first 

 letters of the alphabet, .as a, b, c, d, &c. 

 and the variable or flowing quantities by 

 the last letters, as T>, w, x, y. z : thus, the 

 diameter of a given circle may be de- 

 noted by ; and the sine of any arch 

 thereof, considered as variable, *by x. 

 The fluxion of a quantity, represented by 

 a single letter, is exp : essed b> the same 

 letter with a dot or full point over it : 



thus, the fluxion ot x is represented by 

 x, and that of y by y. And, because 

 these fluxions are themselves often va-^ 

 riable quantities, the velocities with, 

 which they either increase or decrease 

 are the fluxions of the former fluxions, 

 which may be called second fluxions, 

 and are denoted by the same letters 

 with two dots over them, and so on to 

 the third, fourth, &c. fluxions. The 

 whole doctrine of fluxions consists in 

 solving the two following problems, viz. 

 F;om the fluent, or variable flowing 

 quantity given, to find the fluxion; 

 which constitutes what is called the di- 

 rect method of fluxions. 2. From the 

 fluxion given, to find the fluent, or flow- 

 ing quantity ; which makes the inverse 

 method of fluxions. 



FLUXIONS, direct method of, the doctrine 

 of this part of fluxions is comprised in 

 these rules. 



1. To find the fluxion of any simple va- 

 riable quantity the rule is to place a dot 

 over it : thus, the fluxion of x is a'-, and 

 ofy,y. Again, the fluxion of the com- 

 pound quantity .r-f-?/, is x-\-y . also the 

 <fluxk)n of x y, is x y 



2 To find the fluxion of any given 

 power of a variable quantity, multiply the 

 fluxion of the root by the exponent of the 

 power, and the product by that power of 

 the same root, whose exponent is less by 

 unity than the given exponent This rule 

 is expressed more briefly, in algebraical 



characters, by n x x = the fluxion of 



n 



x . Thus the fluxion of x3 is xX 3 X #* 

 = .3 a: 1 j; ; and the fluxion of.rS is x X 5 

 X x* = 5 .14 jr. in the same manner the 

 fluxion of tH^JVi is 7 i/ X oT? 6 ; for the 

 quantity a being constant, y is the true 

 fluxion of the root a -f y Again, the flux, 

 ion of *H-?)f will be -|- x 2 z z X 

 B*'4-z^]|: for here x being put = a 1 -f-z i , 



we have x = 2 z z; and therefore f A- 

 for the fluxion of x f (or +zi)|) is= 3 



3 To find the fluxion of the product of 

 several variable quantities, multiply the 

 fluxion of each by .he product of the rest 

 of the quantities; and tiie sum of the pro- 

 ducts, thus arising, will be the fluxion 

 sought. Thus, the fluxion of x y is xy~$- 

 y .r; that of x y z is x y z -j- y x z -f- z x tfc 

 and that of i> x ?/ z is t x y z-\- x v y z -f- 



