106 DIFFERENTIAL AND INTEGRAL CALCULUS. 



NP be produced to p so that Np = PG, then the area of 

 the surface generated by the revolution of the curve about 

 the axis of x = 2?r x the corresponding area of the curve 

 traced out by p. 



(* X X 



If the curve be the catenary y = - (s~ + s"c), the point p 

 will lie on a catenary y = - + - (s~ -f e~ ^). 



Contact of Curves. 



If (or, y) be a point P (fig. 30) common to two curves, and 

 if when we increase x to x + Bx the corresponding values 

 of y in the two curves are y + ^y^y-\- %', then the equations 

 of the two being y =/(#), ?/ = </>(#), we have 



1S 



By' = $ (a) + 0' (a) S* -f ...+ < n 



= < (x) since the point is common to both curves ; 

 also they will touch each other at the point if/' (x) = </>' (a?), 

 in which case they are said to have contact of the first 

 order. If also /" (x) = <f>" (x)...f rt (x) =<f> n (x}, the two 

 curves are said to have a contact of the n ib order. In such 

 a case, if OM=x, MP=y, ON=x+$x, 



i.e. if we call MN a small quantity of the first order of 

 smallness, QQ' is a small quantity of the (w + l) th order; 



00' 

 i. e. TifXTB+i tends to a finite limit when Bx is made = 0. If 



NQ Q' meet the tangent at P in R, then 



