Mr Sang*s Analysts of the Vibralio7i of Wires. 309 



istence of any regular investigation of their nature, I submitted 

 them to a strict analysis, in the course of which I was led to a 

 variety of interesting conclusions. Induced by the results of this 

 investigation to reject the round wire and to substitute rectangu- 

 lar prisms of various degrees of flatness, I was delighted at obtain- 

 ing, both theoretically and experimentally, many algebraic curves 

 remarkable ahke for their beauty and for their curious properties. 



If a slender steel bar, fixed at one end, and having a small 

 pohshed ball attached to the other, be thrown into a state of vi- 

 bration, the image of a luminous object, seen by reflection from 

 the surface of the ball, is observed to describe certain beautiful 

 curves, which either lie nearly in one plane, or are curves of 

 double curvature, according as the wire is straight or bent. I 

 shall proceed first to consider the motions of the straight wire. 



Let the origin of co-ordinates be placed at the fixed extremity 

 of the wire, the axis z coinciding with the position of rest; and 

 suppose that the free 

 extremity is drawn a- 

 side from ;2 by a dis- 

 tance e, measured in a 

 direction making with 

 X the angle ^. IfEF 

 represent a transverse 

 section of the wire, O 

 being its centre of po- 

 sition OX and OY pa- 

 rallels to the axes X 

 and Y, and OG the di- 

 rection of deflection : OH will be the axis of torsion of that sec- 

 tion of the wire. 



I being any element of the section, the compression upon it 

 will, clearly, be proportional to e . 6z .IK; 6z being some func- 

 tion of z depending on the nature of the curve of flexure, and 

 constant throughout the section EF. The force evolved by this 

 compression will thus be proportional to e . 6z . IK. . ds, ds 

 being the element of the section. Now this force, acting at the 

 extremity of the lever K I, tends to rectify the position of the 

 wire by turning it round the axis OH ; and again conceived as 

 acting at the extremity of the le\'er I L, it tends to turn the 



