Application of the Fluxional Ratio, &c. 97 
a binomial root, in which z is the original fluent, and 2! its increment, 
equal to the fluxional base 2, then if x+-a’ be raised to any assign- 
able power, the result will be equal to the sum of that original fluent, 
its first fluxion, half of its second fluxion, one sixth of its third flux- 
ion, one twenty-fourth of its fourth fluxion, &c. continued until its 
last fluxion is a constant quantity. 2. Each order of fluxions has as 
many sources of increase, from whence the generating quantities 
commence their’ motion, as there’ are units in the coefficient of its 
uxion. Since fora right understanding of the nature of fluxions, 
much depends on a thorough understanding of these elements, they 
demand an attentive consideration. 
In the second power, the first fluxion zs Fig. 7. 
has two sources of increase, DC, and 
CB, and the second fluxion two, Ce, and 7 
Dn. The two generating lines com- “ 
mence their motion at DC, CB, produ- . 
cing the two parallelograms DnmC, 
CcbB, representing the first fluxion 2rz-,. so 
and the two generating lines Cc, Dn, 
commence their motion at Ce, Dn, ape the two squares Cerm, 
Dnwd, representing the second fluxion 22° 
_ The manner in which the several filers of fluxions arise in the 
third power, is made plain by the diagrams annexed to the article, 
page 330 in the xiv.-Vol. of the Journal of Science, to which the 
reader is referred. First fluxions are there designated by the short 
prisms of a red color, second fluxions by the prisms of a yellow color, 
and third fluxions by the cubes of a blue color. ‘The three genera- 
ting squares are described as commencing their motion at the bases 
of the three pyramids, which compose the fluent, forming the three 
short prisms of a red color, whose thickness is z*. These prisms 
represent the first fluxion 3r*2z°. Nextly, the six generating paral- 
lelograms, whose length is equal to a side of the generating squares 
just mentioned, and width equal to z°, commence their motion from 
the two flowing sides in each of the short prisms, and produce the 
six quadrangular prisms of a yellow color, representing the second 
fluxion 6rz-*. Lastly, the six generating squares, whose sides are 
each equal to z-, commence their motion at the ends of the six prisms 
of a yellow color, which are supposed to flow, and to produce the 
six Be of a blue color, —— the third fluxion 62°. 
13 
L. KAV—No. 1. . 
