Division of Vibrating Bodies. 205 



other rectangularly, and I shall examine in what way this 

 figure may pass to another composed simply of parallel lines. 



For example, I suppose that we have produced the mode 

 of division represented in No. 1. Fig. 1, of Plate V. then if 

 the tension of the membrane is constant, its sound becomes a 

 little more acute. It may happen that the angles opposite to the 

 summit in a a', bb', cc', dd', may disunite, as in No. 2, which 

 will gradually assume the aspect of Nos. 3, 4, and 5, (if the 

 sound always ascends,) and afterwards that of No. 6, compos- 

 ed of only four parallel lines. The same transformation may 

 take place as in Fig. 2 and 3. It may also happen, as in Fig. 4, 

 that the opposite angles in a a r , b &', c c', d d', are those which 

 divide themselves first, and that the figure formed by the sand 

 assumes successively the aspects of Nos. 2, 3, 4, 5 and 6; or this 

 division may take place, as in the Nos. 2 of Fig. 5 and 6, which 

 will produce still new modifications in the successive figures., 

 which terminate in four parallel lines. In short, it may even 

 happen that the opposite angles do not divide themselves as in 

 No. 1. of Fig. 7, which passes to that of No. 6, by the simple 

 inflections of right lines in opposite directions. 



A number of parallel lines may pass into another number of 

 lines, either parallel or directed rectangularly. Fig. 8. shows 

 a transformation of this mode of division to two nodal parallel 

 lines, and Fig. 9. a passage from the same mode of division to 

 four lines also parallel, but cut rectangularly by two other 

 straight lines. 



In general, when we set out from a figure composed of rec- 

 tangular lines, the character of the successive modifications 

 depends on the manner in which the angles opposite to the 

 summit divide themselves. This may be observed in a very 

 distinct manner in Figs. 10 and 11, which are the passages of 

 four parallel lines, cut by two other right lines, to six parallel 

 lines. On the other hand, if we set out from parallel lines, we 

 may say in general that the character of the modifications de- 

 pends on the different inflexions which those lines may affect. 

 It is thus that in Figs. 10 and 11, the Nos. 5, considered 

 as the first modifications of the right lines, ought to produce 

 quite different phenomena, dependent on this, that one of the 

 lines bends itself outwards, whilst in the other they bend them- 



