Application of the Fluxional Ratio, &fc 



307 



by their respective ratios, the series will become 



2a 2 x 2 

 3 



x 



s 



"2" 



5a* 



x 



7 

 2 



X 



9 



2 



1_1 



5# 2 



28a* 72a' 704a* 



&c. Let the terms which compose the 



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



then B&cF 



A : : OadE 



L 1 



2a' J x 2 



BbcF : -B::Oat7E { 



BbcF: 



OadE 



BbcF : - D : : OadE 



hence (E.12.5.)B6cF : A 



x 



7 

 2 



X 



9 

 2 



» 



3 



5 







► ._ 



X 2 







* • • • T 



5a 3 



7 









# 2 



• 





♦ 











28a 3 











9. 





• 





X 2 





♦ 













72a* 



i 







1 2 



a: 3 



-B-C 



-D : : OadE 



J 2a 2 x 2 - 



i 









5a 3 



28a 2 



72a^ 



&c.=DOE. 



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 z. 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 



A 



FDc Em 



C 



