DYNAMICS OF A POINT. 



- \ ' : ; r -v. that when it again moves hori- 

 zontally, its vertical depth below the highest point of the sphere 

 is a mean proportional between the radius and its initial distance 

 from tlu 1 lowest point. 



1415. A particle in motion on a smooth surface z=<f>(x, y) 

 under the action of gravity describes a curve in a horizontal plane: 

 if u be its velocity, prove that 



u* ftl'z (<lz\* 



~ 



V \4y dy dy* dm 



\is of z being vertical. 



1116. In a smooth surface of revolution whose axis is verti- 

 cal, a heavy particle is projected so as to move on the surface and 

 [be nearly a horizontal circle; prove that the time of a 

 al os.-illation is 



y( kr rina ) 



(g (k + 3r cos a sin" a)) ' 



k being th- C8 from the axis, r the radius of curvature of 



the g id a the inclination of tl!- t> the 



il, in the mean position of the particle. 



1117. A particle slides in a vertical piano down a rough 

 Mai arc wh..s,. ;ixi ioal, Btarting fr'iu tlie cusp :m<l 



coniini,' t> i-c-t at the vertex; prove that th- r.M-ilicirnt of friction 

 equation 



1418. Three equal and similar p:irti-h'.s ivpi-llin.ix ';ch 



forces varyiog as the distance are connected by equal incx- 



le strings and are at rest : if one of the strings be cut, the 



subsequent angular velocity of fit IP r of the other strings will 



//I 2 cos B\ 

 vary ./(., ^^ )> being the angle between them, 



