BOOK <>F MATlir.MATICAL PROBLEMS. 



Two equal particles are connected by an. in extensible 

 : ; one lies on a smooth horizontal table, and the string passes 

 through a small fixed ring in the edge of the table to the other 

 which is vertically below the ring : the former particle is pro- 

 l on the table in a direction at right angles to the string 



with a velocity ./ - ^-p- , c being its distance from the ring; 

 it 11 (TI + 1 ) 



prove that its next apsidal distance will be - , and that its ve- 

 locity will then be to its initial velocity as n : 1. 



Prove also that the radius of curvature of the projected parti- 



4c 

 clc's initial path is - ^ . 



1428. Two particles, whose masses are p, q, are connected by 

 a fine string passing through a small fixed ring ; p hangs verti- 

 cally, and q is projected so as to describe a path which is nearly a 

 horizontal circle; prove that the distance of p at any time from 

 its mean position will be 



A sin (mt + E) + A' sin (nt + B') ; 

 m, n being the positive roots of the equation 



(af _??(!_ cos a)) L* -- g (1 + 3 cos 2 a)) = ^ cos a (1 - cos a); 



^ C C COS ft j C 



where c is the mean distance of q from the ring, and p = q cos a. 



1429. A particle is placed in a rough tube (fi = -j which 



revolves uniformly in one plane about one extremity, and no 

 forces but the pressures of the tube are in operation ; prove that 

 the equation of the particle's path is 



1430. A rectilinear tube inclined at an angle a to the verti- 

 cal revolves with uniform angular velocity u> about a vertical 



