of Consonances of the Form h : 1. 501 



being smoothed out. The sole effect of the term involving X 

 is in this case to modify slightly the positions of the vertices. 



84. If, then, pX were exactly = qX f , then after a certain dis- 

 tance, which may be called a short cycle, the vertices would 

 recur for precisely the same values of y. And the correspond- 

 ing vertices in successive short cycles would lie on q straight 

 lines, or on 2q straight lines if the lower vertices be included. 



This short cycle is obviously pX=qX' in duration. 



85. Since, however, in our general case pX + 8 = qX r , the 

 coincidence after the short cycle is not exact ; but the vertex 

 determined by equating the second term of the inclination to 

 zero has, in the first term, a different correction to the value 

 of y from that which existed before the short cycle. 



86. At the vertex before the short cycle let x=ct, so that 

 the second term of the inclination vanishes ; then, before the 

 short cycle, 



y = Ecos?p + F; 

 after one short cycle, tV=qX' -\-cc=pX -f- 8 + a, 



^=Ecos^(a + 8) + F; 

 after two short cycles, iv=2qX' + a = 2(p\ + £) + «, 



2«7T 



y 2 =Ecos~0 + 2S) + F; 



and so on, till after n short cycles, where n$ = X, nearly or 

 exactly, 



*/ n =Ecos-^- + F; 



and the ordinate of the vertex in question has gone through 

 a complete period of a pendulum-curve in the space 



nqX r = n{pX + h) 



=7iX(j9+ - j, since 7iS=\ 



==\(wp + l). 



87. Now we have seen that there are q of these vertices, 

 each of which gives rise to one of these curves. Consequently 

 this space, (np + l)\ = nq\', presents, both above and below, 

 q projecting bows, and each bow is of length 



np + 1 



— X or nK\ 



