32 



But it is necessaiy to eliminate v from equation 8. 

 Now — = lim. — = Force in direction of the Tangent 



d t A ^ 



„ ,. PT 7,,. QT _ n dv r, dx 



= P hm. -p-^ - R hm. jtq - P^Ta-p-^Tap 



dAP 



Therefore because v = —^ 



vdv = Pxd f — R dx 

 or by integration 



v^ = J 2 Px dv — / 2 Rdx + k, k being a constant quantity (jj) 

 Hence equation (8) becomes 



dx 



d 



V 



\ '2h + ; t X ' dy x' ^ 



This equation of first fluxions is not so convenient for integrating 

 by approximation as the equation of second fluxions which can 

 easily be deduced from it, as follows. Both, however, will be 

 used in the subsequent investigations- 

 Substituting n = — and making y = f Pi~ dv+fRic~ du + \ k 



and w = 2h + f P u~ d ►, we obtain by squaring, &c. 

 differencing this equation, making d v constant 



dy y d r. ^^ _ d_^ ^jQ 



2 VI d u ^u- uu d-, 



but by equat. (9) 



,/di« " '''■1 "in'^ „\ d to 



tu via u va»" y 



V)d u 



therefore by equation (10). 



1 /dy / du'^ , „\ dw\ rf*« /1l\ 



