NEWTON S PRINCIPIA. 231 



stant factors, the resistance on the curve will be propor 

 tional to 



Then the general equation given by the Calculus of Varia 

 tions leads at once to 



this, therefore, is the differential equation to the curve 

 which generates the surface of least resistance. The con 

 struction which Newton gives agrees exactly with the 

 above equation. 



This equation cannot be integrated ; we cannot, there 

 fore, find the equation to the curve. If we could, we 

 should have two constants in it, which are to be determined 

 by the conditions, first, that when x = o, y = a, and 

 when x = I, y = b, where I is the length of the solid, 

 a and b the radii of the bounding sections. 



The equation to the surface of least resistance may be 

 put under a more convenient form. Differentiating it we 

 have 



p*- 2p z - 3 

 dy = pdx = c -j - dp 



fl 2 *\ ^ 

 . . d x = c L s ~ -3 I dp. 



\p p 3 p 5 J ^ 



Hence, 



a + * c = l og . p + l 2 + JU 



(1 + p2)2 ( 



p 3 ^ 



By eliminating p the equation to the generating curve 

 may be found. 



If we wish to find where the curve cuts the axis of x, 

 we have merely to put y o ; this leads to 



Q 4 



