216 Mr. W. Sutherland on a 



The rod is either loaded or not loaded at the free end according 

 to convenience. Kupffer's error came in in the method in 

 which he tried to eliminate the influence of gravity. Of 

 course, if the rods were made to vibrate in a horizontal instead 

 of a vertical plane, gravity would have no appreciable effect if 

 the rods were stiff enough and unloaded, but the bending effect 

 of a load must needs interfere ; and in the case of lead and 

 tin wires, for example, even when unloaded, it is impossible to 

 make them vibrate in a horizontal plane. A lead wire which 

 will collapse at once if held horizontally at one end, will stand 

 vertically if clamped at the bottom and execute vibrations 

 without collapse ; and even at temperatures at which, when 

 the bottom is clamped, vibrations become impossible through 

 instability, it is always possible, with the top clamped, to get 

 stable vibrations. 



Thus, to get values of Young's modulus by KupfFer's 

 method at any temperature requires only a correct method of 

 allowing for the action of gravity in affecting the period of 

 vibration. 



The advantages of Kupffer's method are several : it can be 

 applied to convenient sizes of material, while the static method, 

 to give accurate results, necessitates the use of long wires; by 

 loading the end of the vibrating piece its period can be made 

 such as is easily measured without special recording-appli- 

 ances, and is obtainable by a mere counting of vibrations in 

 a measured time. If desired, the method could easily be used 

 for determining the influence of stress on the value of Young's 

 modulus ; by loading the free end, the value of the modulus 

 could be found under stresses right up to the breaking load. 



Zoppritz (Pogg. Ann. cxxviii.) has given the theory of the 

 effect of gravity on unloaded bars, but the case of loaded bars 

 has not, so far as I am aware, been solved in a form suitable 

 for practical use ; and as freedom to load at will the specimen 

 under experiment is one of the best points about the method, 

 it will be as well to give first the theory of the lateral vibra- 

 tion of a loaded bar under gravity. 



A slight adaptation of a method of calculation given by 

 Rayleigh in his i Theory of Sound,' vol. i. chapter 8, gives a 

 close enough solution for practical purposes. He shows that 

 the period of a laterally-vibrating bar can be got approxi- 

 mately by supposing the bar bent statically by a deflecting 

 force applied to the free end and then allowed to move, so that 

 at any time t from the moment of release the deflexion of any 

 point on the bar is to its original deflexion in the ratio 

 cos 2irnt to unity. Then, by equating the maximum value of 

 the potential energy during the motion to the maximum value 



