( 328 ) 



CHAPTER XXIII. 



DIFFERENTIAL COEFFICIENTS OF AN ARC, AN AREA, 

 A VOLUME, AND A SURFACE. 



307. THE length of the arc of a curve APQ, reckoned 

 from any fixed point A to the 

 point P, is evidently a func- 

 tion of the abscissa x of the 

 point P. This function is 

 often very difficult to deter- 

 mine, but its differential co- 

 efficient with respect to x can 

 always be assigned. 



Let P, Q, be two points on a curve ; 

 x, y, the co-ordinates of P ; 

 x + A#, y + Ay, the co-ordinates of Q. 



Draw the ordinates PM, QN, and the tangent at P meet- 

 ing QN at R and Ox at T. 



Let AP=s, AQ = s + ks. 



We assume as an axiom, that the length As is greater than 

 the chord. PQ, and less than PR + RQ. 



The chord PQ = V{(Aof + (Ay) 2 ], 



PR = MN sec PTM = MNJ(\ 



-A, 



QR=y+Ay-RN 



= y -f- Ay - (PM+ Az tan PTM) 



