162 VIBRATION OF PLATES. [SECT. xvu. 



The nodal lines and pitch vary not only with the point where 

 the bow is applied, but with the point by which the plate is 

 held, which being at rest necessarily determines the direc- 

 tion of one of the quiescent lines. The forms assumed by the 

 sand in square plates are very numerous, corresponding to all 

 the various modes of vibration. The lines in circular plates 

 are even more remarkable for their symmetry, and upon them 

 the forms assumed by the sand may be classed in three sys- 

 tems. The first is the diametrical system, in which the 

 figures consist of diameters dividing the circumference of the 

 plate into equal parts, each of which is in a different state of 

 vibration from those adjacent. Two diameters, for example, 

 crossing at right angles, divide the circumference into four 

 equal parts ; three diameters divide it into six equal parts ; 

 four divide it into eight, and so on. In a metallic plate, 

 these divisions may amount to thirty-six or forty. The next 

 is the concentric system, where the sand arranges itself in 

 circles, having the same centre with the plate ; and the third 

 is the compound system, where the figures assumed by the 

 sand are compounded of the other two, producing very com- 

 plicated and beautiful forms. Galileo seems to have been the 

 first to notice the points of rest and motion in the sounding- 

 board of a musical instrument ; but to Chladni is due the whole 

 discovery of the symmetrical forms of the nodal lines in vi- 

 brating plates (N. 179). Professor Wheatstone has shown, 

 in a paper read before the Royal Society, in 1833, that all 

 Chladni's figures, and indeed all the nodal figures of vibrat- 

 ing surfaces, result from very simple modes of vibration os- 

 cillating isochronously, and superposed upon each other ; 

 the resulting figure varying with the component modes of 

 vibration, the number of the superpositions, and the angles 

 at which they are superposed. For example, if a square 

 plate be vibrating so as to make the sand arrange itself in 

 straight lines parallel to one side of the plate, and if, in ad- 

 dition to this, such vibrations be excited as would have 



