306 Application of the Fluxional Ratio, S^c. 



term is the fluxion i-'x^x', and the ratio is — =■:; — > hence b^x^x' X 



X 



2&^- 



-Q — =EpGC, the area of the parabola 



In the circles ALH, 

 DNG, (Fig. 4.) let the 

 diameter AH = m, the di- 

 ameter DG=cf, DE=a;, 



mx 



AF= — 5 the fluxional base 

 a 



^d=x', the fluxional base 



7nx' 



i 



Fc= — i'EO={ax-xxy, a fdcE 



FB=-(ax — xx)''^, then the fluxion OadF> = {ax- xx)x', and the 



fluxion 'BbcF=—;^{ax=xxy^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 — xxyx' for its fluxion; 



, nx' ^ 



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-xx) X 



1 1 .1 x'-x- x-x' 



'^ ^' n^^^v 



X^'X' 



^X'^X' 



■a XX' — 



i 



- &c. 



2a^ 8a- IGa^ i28a^ 

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



X 



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



XX XX 



For the first term, — =•;" ; for the second term, — = ^ — ; for 



%X' 



nx' 



\X' 



XX 



the third term, — = :;~, he. If the several terms are multiplied 



nx' 



■kX' 



