dJ8 Mr Sang'^s Analysis of the Vibration of Wires. 



by the exact resemblance between the curves which I had 

 traced on paper, and those exhibited by the motion of the ball, 

 I observed other varieties of the general curve, and found, in 

 the exact agreement of all the phenomena, another confirmation 

 of the generally received law of the elasticity of springs. It is 

 needless to give the analysis of any other varieties; suffice it 

 that I have delineated a few of the more beautiful, and indi- 

 cated the proportions from which they spring. 



Hitherto I have only spoken of vibrations performed by the 

 whole length of the wire, but it is well known that these are 

 frequently accompanied by vibrations of its aliquot parts. The 

 centre of a smaller trajectory, described in less time, is then car- 

 ried along the principal curve. The complex curve thence re- 

 sulting, bears the same relation to these trajectories that the 

 epicycloid bears to the circle ; its general phoronomic equations 

 are, 



/' t — u \ . , f nt — u'\ 

 07 = a cos I ^ \-\-a cosl w — — — j 



, / t — v\ , ,, / 7it — iy\ 

 y — 6cos\^w— — - j + 6 cos\^«r — -~ — J; 



It is not merely possible that the fundamental vibraiions 

 may be accompanied by a set of secondary ones; these also 

 may have their secondaries, and so on ; the equations of the 

 complex curves being obtained by annexing new terms of the 

 forms. 



a cos 



(ni}! t — vl'x J ,^ ( 7in' t — v"\ 

 ^ ^ j and 6" cos (^z^ j 



to the above expressions. 



On account of the great number of arbitraries which enter 

 into such equations, the variety of the curves produced by the 

 same wire is endless. In order to their perfect exhibition, the 

 wires must be much elongated. Those produced by the round 

 wire are by far the most beautiful ; the complexity of the others 

 prevents the eye from catching their entire shape. Were the 

 subject of sufficient importance, it would be easy to examine 

 the nature of these epicycloidal curves. It may already have 

 appeared, indeed, sufficiently trivial for the calculus that has 

 been applied to it ; yet, when it is considered that it is almost 

 the first case of complex vibration which has yielded to a strict 



