Mr Sang's Analysis of the Vibration of Wires. 317 



When the times of vibration of the wire are adjusted in the 

 ratio of 2 to 1 , the general equations of the curve become 



x = acos^^, 2/ = 6cos-A__i; 



when t is eliminated from these, we obtain for the algebraic 

 equation of the curve 



a'{ij + h cos ^)' + h^ {a" - 2 w^ = a^^a^ b* + Ua;'y cos ^) 



which belongs to a continuous curve of the fourth order. 



If the adjustment of the wire be perfect, the form of the 

 curve will continue unaltered until the vibrations die away; 

 but if the adjustment be slightly defective, the line will gra- 

 dually exhibit every form obtained, by supposing v to vary from 

 zero to U. 



On assuming v = 0, the equation becomes 

 a^{b^y) — 2 6^*, 

 which is that of a common parabola passing through A and B, 

 and touching C D at its middle. If we suppose t; == | U, the 

 equation becomes 



a^^i/ = 46^^ (a* — ^«), 

 which is that of a knot of the fourth order, touching AD and 

 BC at their middles, and passing twice through the origin. 

 Lastly, if we suppose v = JJ, the equation is converted into 



a^^b—y) = 25ir*, 

 which belongs to a parabola passing through C and D, and 

 touching AB at its middle. 



The five varieties of Fig. 3. represent the appearance of this 

 curve, when r; = 0, :^ U, i U, | U, and U. 



No circumstance connected with this subject was so nnex- 

 pected as the formation of the common parabola. The object 

 which I first proposed for my amusement was the explanation 

 of the change from the sinistral to the dextral movement, as 

 exhibited by the round wire. Having, for this purpose, inves- 

 tigated the general equations of the motion, I was naturally led 

 to inquire what would be the effects of supposing T and U to 

 bear other ratios to each other than that of equality. The sim- 

 plest of all these, that of 2 : 1 , gave, for the form of the trajec- 

 tory, the common parabola. The hope of seeing this exhibited 

 by the wire itself, led me to employ flat bars. Highly gratified 



