312 Mr Saug's Analysis qfihe Vibration of Wires. 



. But e cos ^ is the distance of the extremity of the wire from 

 the plane Y Z ; ^ sin <p its distance from the plane X Z ; where- 

 fore the tendency of the ball to return to either of these planes 

 is just proportional to its distance from it. The vibrations, 

 therefore, parallel to the directions of greatest and least flexibi- 

 lity are, each of them separately, isochronous, and go on with- 

 out any mutual disturbance. 



Were the elasticity of the wire perfect, the ball would reach, 

 at each vibration in the direction X , to a constant distance from 

 the plane YZ ; and at each vibration in the direction Y, to ano- 

 ther distance, also constant from the plane XZ. The whole 

 trajectory would thus be included in a rectangle, with its sides 

 parallel to the directions of greatest and least flexibility. From 

 the imperfection, however, in the elasticity of all wires, the di- 

 mensions of that rectangle gradually contract. Its shape even 

 is subjected to a variation, depending on the means which have 

 been employed to flatten it : if the wire has been filed, the rec- 

 tangle rapidly becomes elongated in the direction of greatest 

 flexibility ; but if it has first been drawn nearly to the required 

 shape, and then adjusted by hammering, this elongation becomes 

 almost imperceptible. 



If T and U represent the times of vibration, a and b the ex- 

 treme distances of evagation in the directions X and Y, the po- 

 sition of the ball at any instant will be given by the equations 



zit , la (t — v) 

 ^r =: a cos Tpr ; z/ == 6 cos — y? -, 



the time t commencing when the ball is at its greatest distance 

 from the plane YZ, and v being the interval between that 

 epoch and the instant when the ball has reached its greatest dis- 

 tance from XZ. 



The velocity of the trajectile at any instant in the direction 



X is — a -^ sin — ; and its velocity in the direction Y, 

 — b^ sin — ^^^T- — 9 so that its absolute velocity is 



V I T2 U2 J . 



If V be made zero, the beginning of the time t will correspond 

 to the instant when the ball is at the corner of the rectangle;, at 



