232 NEWTON S PRINCIPIA. 





we cannot have c = o, for then y would be always nothing. 

 The numerator of the fraction cannot be nothing, for it is 

 always greater than unity. The denominator cannot be 

 come infinite without making the whole fraction infinite. 

 Hence the curve never cuts the axis of x ; the surface of 

 least resistance has a flat surface exposed to the resistance 

 of the fluid. The next question naturally is, what is the 

 least value of y ? To find this we must put 



i ft 



this gives p = + */ 3 and . . y = + ,- c. At this 



point of the curve there must manifestly be a cusp. 



As p increases from zero, y decreases from infinity ; the 

 curve approaches the axis. The tangent at the cusp lies 

 between the two branches, one going off to p = o, the other 

 to p oo. There are no asymptotes. If y be positive, 

 p is positive. The theory of Equations shows that the 

 equation 



not having more than two changes 

 of sign, cannot have more than two 

 positive roots. Therefore the above 

 two branches contain all that is 

 above the axis of x. If we change 

 r the sign of y, we change the sign 

 of p ; hence the curve is the same 

 on both sides of the axis of x. If 

 we change the sign of c, we shall 

 manifestly have the same curve, 

 except that it is turned the other 

 way, and lies on the opposite side of the axis of y. 



