246 NEWTON S PRINCIPIA. 



where w 2 = ^, and the particle is supposed to be moving 



in the direction in which s is measured. When the particle 

 moves in the opposite direction we must change the sign 

 of x. 



Although the above equation cannot be completely inte 

 grated in finite terms so as to find s in terms of t, yet we 

 can always find the velocity at any point of the arc. The 

 equation can be put under the form 



rf / + 2 l.*=-2*. S 

 as m 



This is a standard form, and the integral will be 

 W 2 6 2* _. _ 2 n* f s . s 2 * 5 ds 



Jd 



where x has been put for . We may find C either in 



terms of the whole arc described, or the velocity at the 

 lowest point. The latter gives when s=o, v = V, 

 hence 



-2* S __ S - - 







This finite expression will always give the velocity. As x 

 is usually small, it may be useful to expand the above in 

 powers of x . We have 



