Application of the Fluxional Ratio, 8fC. 



305 



equivalent to B"x" X 



nx' 



stituent parts. Let APBD, 

 EpGC, be two similar para- 

 bolas, the parameter PF=a, 

 the parameter pF = 6, the 

 abscissa EC = a;, the abscissa 

 AD=mx, then the ordinate 



the fluxion may be resolved into its con- 

 Fig. 3. 



CG: 



■ b X , and the ordinate 





the fluxional 

 the fluxional A 



base Cc=x', the fluxional A E 



base 'Dd=mx'. Since x' and mx- are supposed to be generated 

 contemporaneously, and the ratio depends upon their relative, and 

 not upon their absolute magnitude, we may assign to x' any magni- 

 tude at pleasure. Conceive the two parabolas to be generated by 

 the lines CG, DB moving uniformly from the points A, E, toward 

 c, d. The parabolic spaces EGC, ABD increase with an accelera- 

 ted motion, but the increments are continually diminishing, and the 

 motion approximates toward a uniform motion. Suppose the two 

 fluents to be taken when the generating lines arrive at the points 

 C, D. By the definition of a fluxion before given, the increments 

 are annihilated at the moment the generating lines arrive at C, D, 

 and the parabolic spaces are left to increase uniformly. Hence the 

 fluxions are represented by the rectilineal parallelograms ^ndD= 



a^nfix^x', and GacC=b^x''^x'. Then according to the foregoing 

 proposition, 



aK^x^x- : APBD r.b^x^x- : EpGC. 

 Here, although an expression for the first and third terms are ob- 

 tained, yet the second term is not given, hence the fourth term can- 

 not be obtained by the common rule,, as is the case in trigonometry. 

 But the ratio of the second term to the first can be had. As in trigo- 



nometry the ratio is -» so in fluxions, the general formula of the ra- 



X 



tio is — ■/ The great advantage of this ratio consists in this, that by 



multiplying the third term in the proportion by it, we arrive at the 

 same result as by multiplying the second term by the third, and di- 

 viding the product by the first term. Now, in this example, the third 



