306 



Application of the Fluxional Ratio, fyc. 



1 1 x X 11 



term is the fluxion b"*x*x\ and the ratio is — =- — * hence b^x^x* X 



nx m £x* 



x 



\x* 



3 



3 r* 

 2 X 



EpGC, the area of the parabola. 



In the circles ALH, 

 DNG, (Fig- 4.) let the 

 diameter AH=tw, the di- 



ameter DG=a, DE 



x 



AF 



mx 



a 



j 



the fluxional base 



EdJ=;r, the fluxional base 

 Fc=—t'EOsz{ax-xx) a t A FDcEd 



c 



G 



I 



FB=-(ax— xx) , then the fluxion OadE= (ax- xx) x^ and the 



VHP 



fluxion B6cF=— (as=zx)*x\ It has just been stated, that the 



fluxion multiplied by the ratio will give the correspondent fluent 

 sought. But the analysis requires some fluent, out of which the 

 given fluxion has arisen, which by a contrary process is again made 



. nx* 



to appear. To this end it is necessary, that the ratio — should be 



contained in the fluxional quantity, that the quantities represented by 

 it may be eliminated by the multiplication of its reciprocal. But 



there is no quantity, which will produce (ax—xx 



* 



hence the ratio — is not contained in it, and its fluent under its 



x 



present form cannot be found. We are, therefore, under the ne- 

 cessity of transforming it into an infinite series. It will then be 



(ax-zz) x'=a * v - 



| 5 1 



X-X' x 2 X m x 2 x- 



2a 



I 



8. 



8a 2 



16a* 



9 



5x 2 x* 



128a* 



he. 



It is manifest, that this series presents as many distinct problems, 



x 



requring different modifications of the ratio — , as there are terms. 



For the first term, 



x 

 nx m 



X X 



; j for the second term, 



x 



ix 



nx m 



fa? 



;: for 



xx 

 the third term, — =— , &c. If the several terms are multiplied 



