of Consonances of the Form h : 1. 505 



almost a straight line in the short space occupied by a wave 

 of the higher curve ; and under these circumstances the ver- 

 tices of the higher curve continue to be visible wherever they 

 come upon the vertex of the lower, especially where the two 

 vertices are turned opposite ways. 



The conditions of case I. not being strictly fulfilled, the 

 consequences there deduced do not strictly follow. The con- 

 siderations as to the number of different vertices which develop 

 curves are not materially affected. And it remains true that 

 there are always p curves (in case I. q curves) actually deve- 

 loped ; but it is not true that there are no traces of any of the 

 other q curves of the entire external set of p + q of case II. 

 On the contrary, it is seen in several of the illustrations, where 

 for the most part p=l, that, instead of the outline being one 

 pendulum-curve embracing the outlines of all the Smith's beats, 

 the internal vertices of the long curves present traces of the 

 crossing of two pendulum-curves of longer period — an effect 

 which is seen to survive from the more general cases, on com- 

 paring the illustrations to case II. As the amplitude of the 

 higher note diminishes, this curve assumes a trochoidal form, 

 the external vertices being less sharp than the internal, where 

 there is the survival from the crossing. Ultimately, no doubt, 

 the outline would become theoretically a pendulum-curve. 



"CI 



But, in the case of indefinite diminution of the coefficient ~. 



a F P 



where - is great, ^ is of the order of the product of two small 



quantities ; consequently the effects on the displacements, or 

 the curves we are examining, would themselves tend to become 

 evanescent before their peculiarities ; consequently the curve 

 enveloping the Smith's beats would never in this way be 

 reduced to a pendulum-curve having the period of those beats. 



In the application of these considerations we have, further, 

 to remember that the resultant tones which present pendulum- 

 curves having the periods of Smith's beats are only heard 

 when both notes are pretty loud ; and under these circum- 

 stances the indefinite diminution of the ratio above supposed 

 is not admissible. The only case, therefore, in which a locus 

 of vertices is a pendulum-curve of the same complete period 

 as the period of Smith's beat, is that of an internal system 

 under case II., where q— p = l. As the existence of this 

 system depends on the accurate adjustment of the coefficients 

 to the law pEi = q¥ } it cannot be referred to even as an illus- 

 tration of a phenomenon of general occurrence. 



97. We conclude, in conformity with the explanation at the 

 end of the former part, (1) that the forms exhibited by the 



