( 336 ) 



CHAPTER XXIV. 



CONTACT. CURVATUEE. EVOLUTES AND INVOLUTES. 



316. LET y = <f>(x) be the equation to one curve, and 

 y = TJr(z) the equation to another ; then if < (a) = ty (a) the 

 curves intersect at the point whose abscissa is a. If more- 

 over <f> (a) = -//() the curves have a common tangent at the 

 common point ; in this case they are said to have a contact 

 of the first order. If moreover <j>"(a) =ty" (a) the curves are 

 said to have a contact of the second order. If <f> (a) = ty (a), 

 <' (a) = aj/ (a), <" (a) = -ty" (a) , $'" (a) = ^r" (a), and so on up to 

 <(>" (a) = i|r n (a) , the curves are said to have a contact of the 

 ?i th order at the common point. When we speak of two curves 

 having contact of the n th order we imply that they have not 

 contact of a higher order ; that is, with the preceding notation 

 \ve imply that <"""(") is not equal to -\Jr n+1 (a). 



317. If two curves have at any point a contact of the 

 w th order, then in the vicinity of the common point no curve 

 can pass between them unless it has with both of them a 

 contact of an order not lower than the rt th . For let y = <$> (x) 

 and y = -fy (x) be the equations to two curves which have 

 contact of the w th order at the point x = a ; and let y l denote 

 the ordinate in the former curve corresponding to the abscissa 

 a + h, and y t the ordinate in the latter curve corresponding to 

 the same abscissa ; then, by Art. 92, 



