504 Mr. R. H. M. Bosanquet on the Beats 



The zero values of the sine give the p + q vertices of the first 

 set; and the values which make the cosine vanish give the 

 q—p vertices of the other set. 



92. Each of these vertices occupies, as in the former case, 

 its special position in the short cycle q\'=p\, and lies always 

 on a straight line y = constant when such an exact relation 

 holds. 



93. Also, as before, when the above relation is changed into 

 yV= j pX + ^, it may be shown, by examining the successive 

 arguments of the vertices, that they shift their position in 

 successive short cycles, so that they lie on pendulum-curves of 

 long period ; also that the period of these curves is, for the 

 p + q system, p + q times that of the Smith's beat, and for the 

 p — q system, p — q times that of tke Smith's beat. 



94. The curves of both these systems, with the Smith's beats 

 which form part of them, are readily recognized in all those 

 of the pendulum-curve illustrations which approximately satisfy 

 the condition 



pE = q¥ or \'E=XF. 



In the case of the major third, where there are many ver- 

 tices in each short cycle, and the figure of the short cycle is 

 itself complicated, these curves are not easily recognized. It 

 is necessary to mark a set of corresponding vertices in order 

 to recognize the curve in this case. By the time we arrive at 

 the fifth the curves are quite plain. 



The curves of the p + q system are large and bold, extending 

 completely from top to bottom of the illustrations ; each curve 

 comprises the bow of a Smith's beat both above and below. 

 These may be spoken of as the external system. 



The curves of the q—p system are smaller, and lie nearer 

 the axial line of the illustrations. These may be spoken of as 

 the internal system. In the particular case where q—p = 1, 

 such as q = 2, p=T, the internal system exhibits complete 

 periodic curves having the period of the Smith's beat. 



95. Case III., where F is so small that #F is small compared 

 with /?E. This would fall under the argument of the first case, 

 with the signification of the letters reversed. But as we 

 made the convention q >p } there arise some special points of 

 difference. 



96. Where g is much greater than p, as is the case of high 

 harmonics combined with a fundamental, F has to be very 

 small indeed in order that q¥ may be small compared with j!?E. 

 In this case, unless F is almost evanescent, it is not generally 

 true that the only vertices are those of E (the fundamental); 

 for in these cases the vertex of the fundamental curve becomes 



