NEWTON S PRINCIPIA. 203 



But from (1) we have 



d x\~ l jdx 



dx 



dx ~ x f* 



Tt = ~ mj ? d *&amp;gt; 



whence eliminating , we have 



CL 6 



- (4 

 ^ 



dx* V 2 cos 2 

 an equation from which either p or Y may be found when 

 the relation between x and y, which determines the curve 

 is given. 



The velocity at any point of the curve is that due to one- 

 fourth the chord of curvature. For looking at equation 

 (3), the left-hand side is the denominator of the expression 

 for the radius of curvature K, whence 



g&quot;&quot;~ -L j } 

 a s 



Since =- is the cosine of the angle, the normal makes 



cL s 



with the axis of y, the quantity in brackets is one fourth 

 the chord of curvature. Whence the proposition follows. 

 The equation (4) will also enable us to determine the 

 equation to the path when the law of density and the 

 force is given; as an instance take p constant, and Y 

 = g the force of gravity. Then 



V 2 COS 2 a 

 an equation which can be only approximately integrated. 



Newton takes several examples to illustrate his reason 

 ing. For instance, if the path be a semicircle and the 

 force gravity, we have 



