of Consonances of the Form h : 1. 503 



vV=£(\+y), 



V — fC . i 



P 

 and if p : q is in its lowest terras, 



k=p, and v=p + q, 

 p + q is therefore the number of independent vertices arising 

 from these terms. 



90. Another series of values satisfies the condition ; these 

 are as follows (since sin,i'= sin (ir — x) Ac.): — 



x __1 x — u _ \ / — 2a 



\~2 + ~jr> X ~ 2(X'-X) ; 



__ 3 x — a. • 3V — 2a 



^I* 1 ^" 3 ~2(V-\) > 



until 



2v-l #-« _ (2y-l)V-2a 



2 + X' ' ~ 2(X'-X) ' > 



(2v)V=2(X-\0j 

 X'-X 



v = 



And if this be not a whole number, 



(2v)\ / = 2k(\-\ / ), 



and if jo : (/ is in its lowest terms, 



k=p, v=p — q, or q—p } since q>p; 



q—p is therefore the number of independent vertices arising 

 from these terms. 



91. The relation of these different sets of vertices may be 

 otherwise exhibited by putting the expression for the inclina- 

 tion into the form 



. 2ttx , . 2tt , . 

 sin -r— + sin ^y {/c — a) 



X A* 



= 2sin7r{.,(i + i)-^}cos^{.,g-J)+*}=0. 



