Application of the Fluxional Ratio, S^c. 



307 



by their respective ratios, the series will become 



L 1 1 



5a^ 



x^ 



5x 



-^— he. Let the terms which compose the 



28a2 72a2 lOAa^ 



fluent ABF be represented by A, B, C, D, &;c. 



thenBScF: A::Oa^E: 



1 3. 



BbcF: -By.OadE: . 

 BbcF : -Cr.OadE : . 



X- 



X- 

 28a^ 



BbcF : -Dy.OadE 



X' 



■72a- 



hence (E.12.5.)BkF : A-B-C- 



■D y.OadE : 2a' x- -^_ 

 ba^ 



V 



&;c.=DOE. 



28a 2 72a 2 



If it is required to find 

 the length of the arc DO, 

 (Fig. 5.) called the rec- 

 tification of the curve, let 

 CO be represented by a, 

 DE by X, EO by y, and 

 the arc DO by ^. While 

 the point generating the 

 line DE is supposed to 

 move with a uniform motion, the points which generate the line EO 

 and the arc DO move with a retarded motion ; but at the instant in 

 which they arrive at O, the decrements cease to exist, and the gen- 

 erating points are left to proceed on with a uniform motion. A 

 necessary consequence of this change is, that a right lined triangle 

 Ond is generated, in which On represents the fluxion of x, nd the 

 fluxion of y, and Od, which is the tangent of the circle, represents 



