of Consonances of the Form h : 1. 497 



u——e cos 0— /cos ^ 



+ 1/3 { ^ij^-% cos + ?f!+Z. 2 /cos } 



8 1 e z P \ 



+ 4 V V ° 0S 36> + V ° 0S 3 V 



+ l{w GoUe+ w 2COSi(l) ) 



I .^jf cos (20 + 0) cos (26-$) \ 



+ 4W l (p + 2^ + Q>-2?) 2 J 

 , - f cos2(0 + 0) cos2(g--g) ] 



+ * 7e/ I 4Q> + ?) 2 + 4Q>-?) 3 J 



e 3 /| cos (30 + 0) cos (30-0) | 

 + 7 2 I (3p + ?) 2 + (Sp-qf J 



g^ 3 / cos (0 + 30) cos (0- 30 ) V 



+ 7 2 \ (p + Sq) 2 + (i>-3?) 2 J' 



so that there are six summation-tones and six difference-tones 

 produced by direct transformation of the primaries, when the 

 effect of terms up to the fourth order is considered. 



70. The effect of the neglect of n 2 in the denominators of 

 all these terms, is to place the principal development of any 

 term such as the difference-tone p — q at the point where 

 p — q = 0, whereas if the complete solution were retained the 

 condition for the principal development would be 



?i 2 -(p-q) 2 = 0. 



No known phenomenon enables us to distinguish between 

 these two cases. Every thing happens, so far as we know, 

 precisely as if the simpler condition were that which is really 

 important. 



71. If we proceed to terms of higher orders in the same 

 way, we shall always have, in the result of terms of the n + 1th 

 order, the two following terms representing nth difference- 

 tones, which alone are important for our present purpose 



