﻿192 Pseudo-Regular Precession. 



In a state of Steady Motion in a horizontal circle, 



Q = 0, ^=0, K=^ 2 , 



g = V?x d £- =V . NG, v*=^^g® d £ =g . NY? 



if the normal and tangent at P meet the axis Oy in G and V; 



and then with -f- = tan 6, 

 ax 



oscillates between close limits #±/3, 



Q— (ot+ft) sin 2 \mt + {*—/3) cos 2 fynt, Q = m/3 sin mt ; 



and the particle beats mj2ir nutations per second, syn- 

 chronizing with the beat of a simple pendulum of length 

 \—g\m 2 , where 



A ~ VNG + dx 2 ) co * GV + PR' 



where PR is the semi-vertical chord of curvature upward of 

 the profile curve of the surface. 



Then on a cone X = JGV ; and on a sphere or spherical 

 pendulum, PR = NG, and to radius a, 



- =3 cos 6 + sec 6. . 

 A, 



For a profile given by y = cx 7 \ 



d -l=n^, NG = ,^ = -, v 2 = 9 .NV = w, 

 dx x dy ny * J J 



dx 2 x 2 dx 2 n + 2 



Thus on the surface of a free vortex, where the angular 

 impulse xv is constant of all annular elements of liquid of 

 the same volume, x 2 y is constant, and n— — 2, X = co . 



On a motor or bicycle track of this shape, the steering 

 will be easy, and a change of place can be made without 

 difficulty or danger, as with the annular elements of liquids 

 in the vortex volume. 



A start is made with moderate velocity from the circum- 

 ference of the track where the slope is slight, and the car 



