.Application of the Fluxional Ratio, S^c. 97 



a binomial root, in which a? is the original fluent, and x' its increment, 

 equal to the fluxional base x', then i( x-\-x' be raised to any assign- 

 able power, thei 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 

 fluxion. Since for a right understanding of the nature of fluxions, . 

 much depends on a thorough understanding of these elementSj they 

 demand an attentive consideration. 



In the second power, the first fluxion Fig-'7- 



'^ , w n m r 



Ms two sources of increase, DC, and i 



CB, and the second fluxion two, Cc, and j^\ 



Dn. The two generating lines com- 

 mence their motion at DC, CB, produ- 

 cing the two parallelograms DnmC, 

 Cc6B, representing the first fluxion 2xx', 

 and the two generating lines Cc, Dn, 

 commence their motion at Cc, Dn, producing the two squares Ccrm, 

 Dnwd, representing the second fluxion 2a;- 2. 



The manner in which the several orders 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 x'. These prisms 

 represent the first fluxion Sx^x\ Nextly, the six generating paral- 

 lelograms, whose length is equal to a side of the generating squares 

 just mentioned, and width equal to x', 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 Qxx'^. Lastly, the six generating squares, whose sides are 

 each equal to x-, commence their motion at the ends of the six prisms 

 of a yellow color, which are supposed to flow, and to produce the 

 six cubes of a blue color, representing the third fluxion 6x'^. 



Vol. XXV.— No, 1. 13 



