12 CURVE OF 



Cor. 3. When AD is a maximum, 



^ 32 ' ^ 18 ^ 648 ^ 



= p ^ p2 _^ IL p3 nearly, = maximum : 



from which p will be found impossible. 

 Therefore, x and p increase together. 



Cor. 4. Because v= 2 V9.nnV7^^^!^^^^^^^l» 



when V — o ; M A D — m A n = o, orm'.n:'. A D : A w. 



Therefore, in any curve, when m : n y. AD : 

 A «, n will descend from A to n: and it will 

 not pass beyond n ; for, at w, ite velocity = o. 

 Neither will it rest on the curve, at n, because 

 the curve, n A, falls below the inclined plane 

 w A, on which n would rest : the weight, n, 

 will, therefore, ascend to A. For the same 

 reasons, it will descend again from A. And, 

 therefore, if the cord A w, be perfectly flexible, 

 and if the pulley, at A, move without friction, 

 and if the curve A /^, be perfectly smooth, the 

 weigVit, w, will oscillate between A and n for 

 ever. 



Cor. 5. When m : « :: 1 : 2 :: x : p, x -^ 



■"2 

 P 3 11 , , 



''2 "^ P "^ "2^ P' + "^ P' nearly, 



from which p will be found impossible. 



