of Consonances of the Form h : 1. 495 



65. In virtue, however, of the preceding considerations 

 concerning the value of n, we may materially simplify the 

 whole process for our present purpose by neglecting the term 

 ri 2 u altogether. The original equation then assumes the form 



d 2 u 



-=-%- = — («M 8 + j8« 3 + . . . ) + E cos pt + F cos (qt — e). 



First approximation, 



u = D~ 2 (E cos pt + F cos (qt — e)) 



/E F ■ \ 



= ~ \.pOOBpt + -2 cos (qt—e) J 



= — (e cos pt +f cos (qt — e)) say. 



Substituting this in the remaining terms, we get 



d 2 u -p, _, . . 



-7-2=.hjCosp£ + Jb cos (qt— e) 



— u(e 2 cos 2 pt +f 2 cos 2 (qt—e) + %ef cos pt cos (qt — e)) 



— fi(e*cospt+f 3 cos 3 (qt — e) + ?>e 2 f cos 2 pt cos (qt — e) 



+ 3^/ 2 cos pt cos 2 (g'^ — e)^ 



— 7(V co^pt +/ 4 eos 4 (gtf — e) + 4te 3 fcos d pt cos (g/ — e) 

 + §e 2 f 2 cos 2 pt cos (§tf — e) + 4?/ 3 cos pt cos (#£ — e) 3 ) 



and u is, to this first approximation with respect to all terms, 

 the integral taken twice with respect to t of the right-hand side 

 of the above equation. 



66. There is, no doubt, a difficulty as to the absolute neglect 

 of the term n 2 u. The effect is to make the vibrating-point 

 apparently rest in a position which is not one of equilibrium. 

 Nevertheless the application of the facts to Helmholtz's hypo- 

 thesis requires this proceeding; and it makes no difference 

 whether it is done finally or at first. I think it very probable 

 that damping terms, depending on the second and higher 

 powers of the velocity, play an important part in the real ex- 

 planation. The source of the terms, however, is of secondary 

 importance in the present state of the question. The point is 

 to show that those resultant sounds which depend on terms of 

 higher orders can become great independently of those which 

 depend on terms of lower orders. 



67. Collecting the terms up to the fourth order, transform- 



2N2 



