— of the Fluxional Ratio, &c. 95 
if: — 
n(n—1)(n—2)z oy 7X 
the fourth fluxion. 
Let 2’ be the increment of z, then if x+-2’ be raney a 
to the several powers, the increment of 
x? will be 222/+-/2 
5? 3x22! 4+-S22!? igs 
ah 4x32! +64? 4/2 +4r2/3 +2/4 i 
In any given power, suppose the first fluxion divided by the inde- 
terminate quantity A, the second fluxion divided by the indeterminate 
quantity B, the third by C, &c. to be equal, each, to the correspond- 
ing term in the increment, 2° being supposed equal to 2’ ; then all the 
orders of fluxions, taken until the variable quantity beccmes constant, — 
will be equal to the whole increment, because all the parts taken to- 
gether are equal to the whole. The values of these unknown coefti- 
cients are found in the following manner 3 suppose z-+-z’ is raised to 
a given power, for instance the third, then the increment will be 32°’ 
+ 3zx/2 -+-2/3.; the first fluxion will be 3z2°z°; the second fluxion, 
3z22°  G6zn"* -6z"3 
6zz** ; and the third fluxion, 62** ; then: : bee S Ee 03 =8242/ 
=n(n—1)(n—2)(n—3)2"- 4a ‘; 
iA. Se Gra‘ 
+3z2/2+-2'%. By supposition —4— =32°z’, and Ra =3z2'?, 
and ooh hence A=1, B=2, Cats pe I a 
4g = Bata 4 Sar! +-z/3, the increment. When z-+2' is raised 
to the fourth power, the increment will be 42°2’+ 6x22’? +-422/%+- 
; 4z%z- 12094°? QD4rz-3 W4ar4 
z's, Proceeding as before 4-+—p—+-GE- +-p-= 
. 3 4 Zip 
4x°7'+-6272/? +-477/3-+-2/4. By supposition ee =4z'z', hence 
12724°2 2472 ‘ 
A=1; B= 62?2'?, hence B#2; oh =417'/*, hence C= 
2424 : 4n3x- \ 12¢%¢2 
6; D =2/4, hence D=24, Therefore 5 aes oa 
24xt? QAx4 
“Tago haa =42732/+6222/2 +4r273-+-2/*,the increment. To 
avoid the difficulty of indicating by points a fluxion of a very high 
