FLUXIONS, 



WiMe kinds, is to find their fluents by ap- 

 proximation, which, by the method of in- 

 finite series, may be done to any degree 

 of exactness. See SEIULS. 



Thus, if it were proposed to find the 



fluent of , it becomes necessarv to 



a x 

 throw the Muxion into an infinite series, b) 



dividing a x by a x : thus, a x-r- 



Now the fluent of each term of this se- 

 ries may be found by the foregoing 1 



rules to be x 

 -K&e. 



- . . 



+ + _ -f _-f 



5 a* 



In order to show the usefulness of 

 fluxions, we shall give an exam .le or two, 

 1. Suppose it were required to divide any 

 given right line A B into two such parts, 

 \A C, C B, that their products, or rec- 

 n^ngles may be the greatest possible. 

 Let A B = a, and let the part A C con- 

 sidered as variable (by the motion of C 

 towards B) be denoted by x. Then B C 

 being =** a x, we have" A C X B C = 

 a x -r- x x f whose fluxion ax 2 x x be- 

 ing put = 0, we get a x = 2 x x; and, 

 consequently, x = % a. Hence ii appears 

 that A C (or #) must be exactly one half 

 of AB. 



Ex. 2. To divide a given number a 

 into two parts, *, y> so that x"t yn may be 

 a maximum. 



Since x-\-y =a, and x y n = max. the 

 fluxion of each = 0, the former, because 

 it is constant, and the latter, because it is 

 a maximum : .'. x -\- y -*= 0, and m r/ 

 x m IJiJf-n xm ynl I; = 0: hence, x = 

 nx m ynly nX y 



y. and a- = ------- -= 



y. 



therefore y = 



m y 1 \ 

 n x it 



m y 



m y 



and m : n :: x : y. Now y = - ; .'. x 

 m 



n x ma 



Ji -- = a, consequently x = ; . 



r m ?ra-fn 



nx\ ir a. 



and y( = )== - . 

 V m m-^-n 



If wi=7i, the two parts are equal. 



Cor. Hence, to divide a quantity a 

 into three parts, x, y, z, so that x y z may 

 be a max. tbe parts must be equal. For 



suppose x to remain constant, and y, z, tc> 

 vary ; the product y , and consequent- 

 ly-*' y s > w 'll be the greatest when y = z. 

 Or if y remain constant, the product x r, 

 and consequently y x z, will be great- 

 est when x = z. Thus it appears that 

 the parts must be equal. And in like 

 manner it may be shown, that whatever 

 be the number of parts, they will be 

 equal. 



Ex. 3. Given x -f- y + z = a, and x y" 

 z$ a maximum, to find x, y, z. 



As ,r, y y z, must have some certain de- 

 terminate values to answer these condi- 

 tions, let us suppose such a value of y to 

 remain constant, whilst r and z vary till 

 they answer the conditions, and then x 

 -j- r = and 2? x + 3 x z 2 ~ = 0; hence, 

 ._ _ . 3 x z- z 3xz 



23 z ' 



= 3 x. Now let us suppose the value 

 of Z to remain constant, and x and y to 

 vary, so as to satisfy the conditions - r 

 then x -f y = 0, y 1 x -f 2 x y y = ; 

 2 x if it 2 x i> 

 hence, x = ii = 215 -* - 



. y* y 



.. y = 2 x ; substitute in the given equa- 

 tion, these values of y and z in terms of a:, 

 and a- -f- 2 x -f- 3 x = a, or 6 x == a, 



1 11 



hence, x = - a; .'. y =H; z = ^ a. In 



like manner, whatever he the number of 

 unknown quantities, make any one of 

 them variable with each of the rest, and 

 the values of each in terms of that one 

 quantity will be obtained ; and by substi- 

 tuting the values of each in terms of that 

 one, in the given equation, you will get 

 the valae of that quantity, and thence the 

 values of the others. 



Ex. 4. To inscribe the greatest paral- 

 lelogrthn D F G I, in a given triangle 

 ABC, fig. 10. 



Draw B H perpendicular to A C ; put 

 A C = a, B H == b, B E = x, then E H 

 = b x ; and by similar triangles, b : a 



QOC 



:: x : D F = hence, the area D F G I 

 o 



a x 



= r\- x = max. or a: X x = M 

 b 



b x x 1 = max. .'.b x 2 x x = ; 

 hence, x = b; therefore E H = x-B H. 



Ex. 5. Let ABC represent a cone, 

 A C the diameter of the base to in- 



